ML20206R839
| ML20206R839 | |
| Person / Time | |
|---|---|
| Issue date: | 03/29/1999 |
| From: | Yanez V NRC OFFICE OF THE CONTROLLER |
| To: | Mcknight J NRC OFFICE OF THE CONTROLLER |
| Shared Package | |
| ML20206R845 | List: |
| References | |
| NUDOCS 9905210001 | |
| Download: ML20206R839 (83) | |
Text
{{#Wiki_filter:I p or: p Y UNITED STATES g ) f g NUCLEAR REGULATORY COMMISSION WASHINGTON, D.C. 20555 4001 4 ,o 9***** March 29,1999 NOTE TO: Jim McKnight, RMB/IMD/OClO FROM: Vicki Yanez, PSB/IMD/OClO /j I k
SUBJECT:
NRC SPEECHES, PAPERS, AND JOURNAL ARTICLES Please place the attached speeches, papers, and joumal articles in NUDOCS and the Public Document Room. Thanks. I Attachments: 14 Documents \,
\ \
I l _ , i'O T d' 3 9905210001 990329 PDR ORG NECCN .,y f f-D ,l.Yl 97ogz uon y en(v+2n ' ' J
l 5.. , Invited Tutorial Paper BAYESIAN PARAMETER ESTIMATION IN PROBABILISTIC RISK ASSESSMENT by Nathan O. Siu' and Dana L Kelly2
'U.S. Nuclear Regulatory Commission Office of Nuclear Regulatory Research Washington, D.C. 20555-0001 2
Idaho National Engineering and Environmental Laboratory 11428 Rockville Pike, Suite 410 Rockville,MD 20852 i Submitted to Reliability Engineering and System Safety for publication on October 10,1997. This preprint is not to be cited or reproduced. Abstract Bayesian statistical methods are widely used in pmbabilistic lisk assessment (PRA) because of their ability to provide useful estimates of model parameters when data are sparse and because the subjective probability framework, from which these methods are derived, is a natural framework to address the decision problems motivating PRA. This paper presents a tutorial on Bayesian parameter estimation especially relevant to PRA. It summarizes the philosophy behind these methods, approaches for constructing likelihood functions and prior distributions, some simple but realistic examples, and a variety of cautions and lessons regarding practical applications. References are also provided for more in-depth coverage of various topics. KEYWOR DS: Bayes' Theorem, probabilistic risk assessment, subjective probability l l I l L The views and conclusions in this paper are those of the authors and should not be interpreted as necessarily representing the views or official policies, either expressed or implied, of the U.S. I Nuclear Regulatory Commission or the U.S. Department of Energy. l
1
~
., , . , , : t
-~ .
- 1. INTRODUCTION A probabilistic risk assessment (PRA) is an exercise aimed at estimating the probability and consequences of accidents for the facility or process being studied. Although not required by this definition, PRA often involves the analysis of low-probability events for which litde data are .
available. Bayesian paraineter estimation techniques are useful because, unlike classical statistical methods, they are able to incorporate a wide variety ofinformation types, e.g., expert judgment as well as statistical data, into the estimation process. In addition to their ability to deal with s. parse data, Bayesian techniques are appropriate for use in PRA because they are derived from the framework of subjective probability. This framework, which holds that probability is a subjective (internal) measure of event likelihood,is an integral part of cunent theories on decision making under uncertainty (e.g., [1]). A further, practical advantage of the subjective probability framework in PRA applications is that propagation of uncertainties through complex models is relatively simple. On the other hand, it is very difficult, and intractable in "real" problems, to propagate classical statistical confidence intervals through PRA models to estimate a confidence interval for a composite result of interest (e.g., core damage frequency). For these reasons, Bayesian techniques, first championed for use in PRA in the late '70s [2,3], have become widespread in that field. This paper presents a tutorial on Bayesian techniques especially relevant to PRA. It outlines the motivation behind the Bayesian approach, summarizes the general parameter estimation problem, presents methods for constructing likelihood functions and prior distributions, discusses some useful results, and ends with a variety of cautions and ! comments regarding practical applications. The intent is to provide an expository review with simple but realistic examples. A number of references are provided for readers interested in exploring various topics in greater detail. l 1 l
p;; . ;.;
- 2. MOTIVATION AND FRAMEWORK The following discussion provides an introduction to the issues motivating the use of Bayesian methods in PRA and outlines the parameter estimation problem which is the focus of this paper. More detailed discussions can be found in Refs. 4 and 5. "
2.1 Why Bayesian Estimation? Consider a safety system in an Edustrial process plant. As part of the standard PRA process for computing plant risk, we must estimate the unavailability of the system. This unavailability,in turn, is a function of the unavailability parameters (demand failure probabilities, running failure rates, maintenance intervals, etc.) for the system components. The unavailability parameters are to be estimated using available data. Because equipment failures tend to be relatively rare events, empirical data for parameter estimation are generally sparse. Classical statistical methods are ill-suited for this situation, leading in such cases to excessively wide confidence intervals. Partially because of this, much of the risk assessment community has turned to Bayesian methods (so-called because they employ Bayes' theorem, described below) as a natural means to incorporate a wide variety of forms of information (in addition to statistical data, e.g., "r failures in n tests") into the estimation process. In the Bayesian framework, the analyst's uncertainties in the parameters due to lack of knowledge are expressed via probability distributions. This is a distinct and important departure from the classical (frequentist) philosophy. (In the classical framework, there is no uncertainty in the parameter being estimated;it has a true, albeit unknown value. Confidence intervals are used to quantify variability in the parameter estimators.) But is it legitimate to use information other than empirical data? Isn't such information subjective? Doesn't its use open the door to misapplication and abuses? The short answers to these questions are, respectively, "yes," "yes," and "sometimes". To further expand on these answers, it is useful to make a slight detour to examine the notion of subjective probability and the PRA-based decision making process. 2.2 Subjective Probability and 'Jecision Making As stated earlier, one objective of a PRA is to estimate the probability of accidents for the facility or process being studied. Practical complications arise in this process because there are a number of different interpretations for the term " probability." The most common interpretation is the " classical" or "frequentist" definition, in which probability is held to be, in an infinite series of repeatable, identical experiments (trials), the limiting ratio of successes to trials.' With this interpretation, the probability of an event is directly measurable in principle. A second interpretation is the " subjective" one, in which probability is taken as a quantity that measures the
- 7'e will use " classical' and "frequentist" synonymously in this paper. This cxmforms to the usage of the terms by PRA pactitioners, but not to the usage of steders for whom probability has the classical interpretation (due to Pascal) as the ratio of number of occunences of an event to the number of oppommities for occurrence, where the number of opportunities is finite, and the event is equally likely to occur in all trials.
2 1
1
. \
,.q. .,:;
assessor's degree of cenainty (or uncertainty) as to the truth of a given proposition. (A proposition is a statement which can be either true or false.) Because this probability is based on the assessor's internal state of knowledge, there is, in most practical situations, no "true" or " correct" probability for a given proposition. Indeed, the probability will change as the assessor gains additional relevantinformation. .. In both cases, probability is required to obey the axioms of probability theory and, therefore, the calculus of probabilities. However, the two interpretations lead to diffeirnt degrees of usefulness in various applications. The subjective interpretation is particularly well suited for the decision problems addressed by PRA. To illustrate why this is so, consider the simple decision tree shown in Figure 1. The decision-maker has two decision alternatives, A and B, whose consequences are to some extent uncenain. If the decision maker opts for A, an accident can occur with probability pA IOd consequence Ca. Likewise, if the decision maker opts for B, an accident can occur with probability p, and consequence C,. If the decision maker is to make a "best" decision, p, and p, must reflect the uncertainties in the outcomes due to all recognized sources. Not only must they reflect any uncenainties due to inherent randomness in the process - coin flipping and Brownian motion represent two classic examples of such aleatory (or stochastic) uncertainty, they must also reflect epistemic (or state-of-knowledge) uncertainties, uncertainties that arise due to imperfect knowledge on the part of the decision maker (or the analysts supponing the decision maker). Of the two interpretations of probability we have mentione'd, only the subjective interpretation explicitly allows for the incorporation of epistemic uncertainties. It therefore is the l one better suited for decision analysis applications, such as those driving PRA. 1-PA I A PA CA l 1 - pg j B 4 PB CB Figure 1 -Simple Decision Tree 8 One comunon fannulation of the axioms is, for arbitrary propositions A and B and the cenainly true proposition Q: (i) P{A) = 0 (ii) PtQ) - 1 (E )PtAt U A2 U.4 - PtA )1 + P{A 2) + where the A,'s are mutually disjoint. These are secrred to as the Kolmogorov axioms. 3
6*; **: ,. Let us now return to the three questions that initiated this detour.
+ ls the use of non-statistical information legitimate?
According to the decision analysis perspective, all foims ofinformation that will enable the ... decision maker to make a better decision are legitimate.
- Is non-statistical information subjective?
Non-statistical information can take many forms, including anecdotes, expert opinion, and l model predictions. Some of these forms are more subjective, that is, dependent on the personal knowledge and characteristics of the assessor, than others. It is interesting to note that in practical PRA applications, most information has a degree of subjectivity. Even
- empirical data are subject to interpretation, due to variations among plants and over time (e.g., a significant event generally leads to changes in practices, thereby drawing into question the notion of repeatable experiments), and even ambiguity and incompleteness in event reports [6].
= Does the use of non-statistical information open the door to abuses?
Clearly, abuses are possible when the subjective probability framework is employed. Of course, abuses are certainly possible using empirical data and a frequentist framework, as witnessed by the well known saying about lies, damned lies, and statistics. It is important to recognize, however, that the subjective probability framework does not give the analyst complete freedom. The subjective probabilities am constrained to behave according to the laws of probability, and these can be fairly strong constraints. For example, as will be illustrated later, these constraints force the numerical results of the analysis to converge with the results of classical statistics when there are large amounts of data. (This is a property not shared by other measures of state-of-knowledge uncertainty, for example, the membership functions in fuzzy set theory.) As a practical matter, real-life decision making often uses subjective information; we can strive to ensure that subjective information is used formally and consistently, but we cannot ignore it. With the assertion that " probability" is to be interpreted as a subjective quantity, we can now formulate the problem that will lead back to our main subject: Bayesian parameter estimation. 2.3 Estimation Problem - Framework Consider, as an example, a standby pump. If the pump is demanded, there are two possible outcomes: it will either start or not start. If it is demanded n times, there are 2' possible outcomes (unique sequences of successes and failures), assuming repair follows failure. The outcomes are assumed to be " aleatory"(i.e., driven by chance). (Apostolakis [7] recommends the , use of the terminology " aleatory uncertainties" instead of " random" or " stochastic" uncertainties because the lauer terms are also used in diffemnt contexts.) 4
)
The standard PRA "model of the world" for this situation uses the binomial distribution. This model assumes that the pump responses to the demands are " exchangeable," i.e., the probability of observing r failures in n trials is independent of the orderin which the successes and-failures occur. Using the binomial distribution, the conditional pmbability of observing r failures
' in a trials, given $, the probability of failure on demand, is given by: .
P{r failures in n trialsI$} (1) r !(n- r)!$'(1-$)"' The question is, when Eq. (1) is to be applied, what is the numerical value of the parameter
$7 In principle, $ can be viewed as a property of the pump in the following sense: if we consider a thought experiment in which n increases to infinity, we would expect that r/n tends to a fixed value. (Of course, this is the relative frequency definition of probability.) However, in practice, we don't have an infinite number of trials. Thus, the value of $ is uncertain, and there is no one "right" value of $ which should be used in Eq. (1). Rather, we need to develop a subjective l probability distribution for $, denoted by x($).8 Using the terminology of Apostolakis, this distribution quantifies the "epistemic" (i.e., relating to intellectual knowledge) uncertainties about the true value of $[7].
A similar situation arises when modeling failures while equipment is operating. In this ! case, the standard PRA aleatory model uses the Poisson distribution: l 4 P{r failures in [0,t]I A} - (M)'r! e-M (2) Again, the characteristic parameter of the distribution, the failure rate A in this case, is uncertain and is characterized by an epistemic distribution x(A).* The general framework illustrated by these two examples, in which a model of the world has uncertain parameters O characterized by an epistemic model x(0.), applies broadly to all aspects of PRA. The PRA models of the world typically deal with aleatory events but are not restricted to these; deterministic models for physical phenomenology are also employed. (Note that Apostolakis [4] provides a further generalization of the framework not widely implemented in PRA, in which the uncertainties in the form of the model of the world are explicitly addressed.) The problem of prime interest in this paper concerns the development of x(0). We state it in the form of two questions: 8 To simplify notation in this paper, we will not distinguish between the randorn vanable (often denoted by upper came letters in more formal presentations) and the values taken on by that sandorn variable (often dmoted by lower case letters).
*The widely used" probability of fiequency" PRA framework described by Kaplan and Gamck [5] is best formulated in terms of the Poisson example. This framework distinguishes between parameters CL-r=W") whose values can,in principle be objectively daernmed given sufficient data, and the subjective probabilities of the parameter I values.
l 5
s :: . Given an epistemic model x(0), how should it be modified (updated) when new information is obtained? . How can this model be developed in the first place? Current answers to these questions are the subject of the remainder of this paper. 1 6
.; c ,
- 3. BAYES' THEOREM According to the theory of subjective probability, Bayes' Theorem is the only way a coherent analyst, i.e., an analyst whose probabilities behave according to the laws of probability
[8], can update his/her state of knowledge. This section describes the general features of Bayes'
- Theorem, and introduces a simple application of the theorem. Additional material, both introductory and advanced, can be found in numerous texts (e.g., [9,10]).
3.1 General Features The probability of proposition A, as discussed in the previous section, can be interpreted as a measure of an analyst's beliefin the truth of that proposition. The " conditional probability of A, given B," P{AIB}, measures the analyst's belief that proposition A is true, given (assuming) that proposition B is true. It is defined mathematically, for P(B)
- 0, by
^
P{A I B}- (3) P{B} Note that by definition, both P{A} and P{AIB} are conditioned on the analyst's state of knowledge. In this paper, it is understood that all probabilities are conditioned on this state of knowledge, although wo will not show this conditioning explicitly. Although the subjective interpretation of probability holds probability to be an internal measure, it does not allow arbitrary assignments of values. In order to be meaningful, the analyst's probabilities must obey the laws of probability.' Of particular interest here, they must obey Bayes' Theorem, which follows from Eq. (3): P{AIB} P{Bl A}P{A} (4) J
~
P{B} let A represent a proposition of interest and B represent some new information. P{AIB} denotes the analyst's probability for the truth of A, given new evidence B. Bayes' Theorem then j states that this probability must be proportional to the product of: a) the analyst's probability for the ! truth of A prior to the collection of new evidence, i.e., P{A}, and b) the probability that evidence B would be observed if A is indeed true, i.e., P{BIA}. The proportionality constant is provided
)
by P{B}. This paper deals with the estimation of parameters. Thus, in the case of a single p--octcr 8, the propositions are typically of the form {0 s parameter value s 6 +d0)} and Bayes' Theorem I takes the form
]
- As discussed by Fahaa==n. Slovic, and Tversky [11). without fandhark pronded by consistency tests, analysts' probabilities are often inconsistent and do not obey the laws of probability. The theory of subjective probability is prescriptive,i.e., normative, rather than descriptive;it provides rules that must be followed to ensure the analyst prendes cohesent probabilities that can be usefully employed in rational decision making.
7 3
~
) 1 MEI 0)no(0) zi(01 E)- (5) fL(E10)ao(0)d0 e
Here, x,(0), the analog to P{A} in Eq. (4), is the prior probability density function for the - unknown parameter 0 (prior to obtaining new evidence E). MBO), the analog to P{BIA}, is the likelihood function. It is either the conditional probability of observing E, given 0, or proportional to that probability. The left-hand side of the equation, x (0IE), is the posterior probability density 3 function for 0 after E is obtained. The integral in the denominator, the analog to P{B}, ensures that x (ole) integrates to unity over all possible values of 6 (i.e., that the posterior density function is indeed a proper density function). It is the expectation of 480) with respect to the prior distribution x,(0). Note that E is conventionally in the form of empirical data, but need not be. Other forms of evidence that fit within the framework of Eq. (5) include expert opinions, model predictions, and " fuzzy" or imprecise data. It is extremely important to note that as the amount of evidence increases, the numerical results of Eq. (5) will converge with those of classical statistics. More precisely, the posterior distribution for 6 will become increasingly peak:d about the maximum likelihood estimator for 0, i.e., that value of 6 that maximizes the likelihood function MB0). While we will not prove this result (interested readers can consult a number of texts, e.g., Refs. 9 and 12) we do note that it is intuitive: as we collect more evidence, the information contained in this evidence should overwhelm the information contained in the prior distribution. A demonstration of this result relevant to PRA applications is provided in the followir.g example. 3.2 Example Application i The first step in Bayesian parameter estimation is identification of the pammeter(s) to be estimated. (This involves, among other things, consideration of the form of the likelihood function appropriate to the evidence that will be collected.) The second step is development of a prior distribution that appropriately quantifies the analyst's state of knowledge concerning the unknown parrmeter(s). The third step is collection of evidence and construction of an appropriate likelihood function. The fourth and final step is derivation of the posterior distribution using Bayes' Theorem. The following simple example illustrates the last step. The preceding three steps, whose results are assumed in the example, are discussed in following sections. In this example, we would like to estimate the failure rate (A) per operating hour for a motor-driven pump. Assume that A is constant in time (i.e., there are no aging effects) and that a prior distribution for A has been developed from industry-wide data. Further assume that this
" generic prior" is a gamma distribution, i.e.,
no(A) po-1P(a) e+ (6) 8
and has a mean of 3.0 x 105/hr and a standard deviation of 7.4 x 105/hr. (Methods to develop prior distributions are discussed in Section 5.) For a gamma distribution, the mean and standard deviation are given by E[A] E '- 9 (7) SD[A] J5 Thus, it can be easily shown that a = 0.1644 and = 5478 hr. Assume that the pump is operated for t = 1000 hours without failure. We will first update the prior distribution to calculate the posterior distribution for A. (In this particular case, where the prior distribution is generic, this updating process is referred to as the " specialization" of the generic distribution [13].) We will then compare the results obtained with those from classical analysis.
- Bayesian analysis A constant failure rate implies that failures are generated by a Poisson process. The likelihood function for this evidence is therefore the Poisson distribution shown in Eq. (2): l L(r failures in [0,t]l K)- (At)'r! e-A' (8)
Using Eqs. (6) and (8) in Bayes' Theorem [Eq. (5)], we obtain
'(At)' _u , "A"-' g r! F(a) xi(Al E) . .
,,, (9)
-k -
dA e f, rl F(a) e+ or , xi(AI E) p.a'ga' 1F(a') e-D (10) where a' - a + r
- +t Eq. (10) is derived by recognizing that the numerator in Eq. (9) is of the same functional form with respect to A as Eq. (6), that the denominatorin Eq. (9) is a constant, and that the integrals of both xi(AIE) and n o(A) over all possible values of A must be unity.
9 l l
s
~
s ;;; . In general, for arbitrary combinations of likelihood functions and prior distributions, Eq.(5) must be evaluated numerically. As shown in this example, however, certain special combinations will analytically result in posterior distributions which are of the same form as the prior distribution. The two sets of " conjugate pairs" most commonly encountered in PRA applications are the Poisson likelihood / gamma prior pair used in this example and the binomial ..- likelihood / beta prior. These two pairs are summarized in Table 1. Table 1 - Conjugate Ukelihood-Prior Pairs Most Commonly Used in PRA Applications Ukelihood Function Form Prior Distribution Form Posterior Distribution Form Binomial . Beta Beta n! F(a + p) ._ (, _ )p.: T(a' +p ) r(g_4)n-r g. (3, )p _3 r!(n - r)! T(a)T(p) T(a')T( ') a' - a + r F-p+n-e Poisson Gamma Gamma 0.t)',_u p* Ao-1 p.exe- g,x r! T(a) ,, T(a') a' - a + r s'- p + t Returning to our example and applying the data r = 0 and t = 1000 hours, n' = 0.1644 and p' = 6478 br. From Eqs. (7), the posterior mean value of A is 2.5 x 105/hr and. the posterior standard deviation is 6.3 x 105/hr. Using a spreadsheet program with a built-in inverse gamma j cumulative distribution function or gamma quantile function, it can be shown that the 95% one-sided probabijity interval for A is (0,1.4 x 10"/hr).2 In other words M p.d go'-1 e-#'AdA (11) 0.95 -g f I'(a') d where bs, the 95th percentile for A, equals 1.4 x 10 /hr. Similarly, the 5th percentile of A, b5 equals 1.2 x 10'2/hr. Alternative approaches for evaluating the integral include using currently available equation solving packages, using tables of the incomplete gamma function, or by 8 l A caution in using comrnercial software: make sure of the parametenzation being used for the gamma probability density funcuon. In some programs, the parammer called p is the reciprocal of the parammer used in this paper. 10
transforming the gamma random variable to a chi-square random variable and using published percentage points of the chi-square distribution. In the last case, the product 2p' A follows the X' distribution with parameter 2a': 2 " 2 ' A ~X (2a') (12) - Then X's(2a + 2r) A95" 2( + t) 2 ( } Aos x 3(2a +2r) 2(p +t) The prior and posterior density functions are plotted in Figure 2. It can be seen that the influence of the data is to reduce the probability of large failure rates. However, the data do not j represent strong evidence; the posterior distribution for A is only slightly different from the prior. l
)
{ l
)
3.00E+03 -- I 2.50E+03 -- i Y*' 2.00E+03 -- e g 2 y 1.50E403 -- 1.00E+03 - - 5.00E402 -- l 0.00E+00 l l l l l l l l a 1 n i vi i s i 6 a I I A (hr-1) Figure 2 - Prior and Posterior Distribution for A 11
7,,_.... .. . _ . _ . . - , . - _ . . . . .-
.V .
. Cinctical annivsis The maximum likelihood estimate of A for evidence of the form {r failures in time t} is, for a Poisson process,just ,
$1t (14) in our example, this gives a value of 0. The upper limit of the 95% (approximate) one-sided classical confidence interval for A is, for a Poisson process, given by A,5 ,X*5(2r + 2) (15) 2t where the numerator is the 95th percentile of a chi-square distribution (tabulated in elementary texts on statistics) with 2(r+1) degrees of freedom. (Note that the classical confidence interval is approximate, because the random variable being modeled is discrete. The confidence interval we cite is conservative, meaning that the level of confidence is g.lguii 95%. In our case, r = 0, so we obtain x$5(2) 5.99
- - 3.0 x 10-3/ hr (16)
A95 2(1000hr)2000 hr which is over an order of magnitude higher than the value obtained using the Bayesian approach. It has been pointed out earlier that the results of the Bayesian and classical analyses converge with larger amounts of data. In this example, Eqs. (10) and (13) show that the effect of the prior distribution parameters will decrease as r and t increase. Thus, the posterior mean value of A, a'/$' [see Eq. (7)], approaches the maximum likelihood estimate of A, i.e., r/t [see Eq. (14)]. Furthermore, the posterior standard deviation of A, 8/p', will approach 0. This means that for large amounts of data, the posterior distribution will be highly peaked about the maximum likelihood estimate. Note that while Eqs. (13) and (15) show that the numerical values of the Bayesian and classical 95th percentiles will converge as r and t increase,it should be cautioned that , I Bayesian and classical confidence intervals represent two different concepts. Bayesian confidence intervals express the analyst's subjective beliefs concerning the v6e of A itself, whereas classical confidence levels quantify variability in the estimator of A. Figure 3 shows the impact of increasing r and t (while keeping their ratio constant) on the shape of the posterior distribution. 12
4.00E44 -- Maximum Ukelihood Estimate 3.50E+04 -- l 3.00E44 -- l Mw m2 150E44 -. i [ T=500,000 hr
^
l y 2.00FA04 --
' l r=10 1.50E+04 -- l T=100,000 hr l l
1.ME+04 - T=10,000 hr l 5.00E+03 -- l 0.00E+00 l l , , , Ie i w i i Ia l 1 s I i A (hr-1) Figore 3 - Effect ofIncreasing Data on Posterior Distribution for A l l 13
- 4. THE LIKELIHOOD FUNCTION
*Ihe constmetion of an appropriate likelihood function requires engineering / scientific knowledge specific to the process being modeled, as well as probabilistic modeling expertise. This section raises a number of general points on modeling considerations and provides some functions ,
widely used in PRAs. A special topic relevant to the construction of the likelihood function, i.e., the treatment of uncertain data, is dI; cussed in Section 7. 4.1 Modeling the Data-Generating Process In a standard estimation problem, the analyst is provided with a stream of observed data for a random variable [e.g., x = (x3 ,x2,...,x,,...,x,)is a stream of data for a random variable x] and is asked to use these data to estimate the parameters of the undedying process that generated the data. This requires development of a model for that underlying process. Although there are no comprehensive sets of rules to prescribe "best" models, a few pointers can be helpful in PRA applications.
- 1) Determine if the random variable is discrete, continuous, or mixed. In reliability and PRA
. applications, the most commonly used discrete variable is the number of component failures (or successes). The most commonly used continuous variable is the component failure time.
- 2) If the random variable is discrete, determine if the underlying generating process generates events on a demand basis (i.e., over a number of trials) or over time (e.g., due to modom shocks).
a) The binomial distribution, which treats the Bernoulli (or " coin flip") process, is commonly used in PRA to model processes in which events are generated on a demand basis. This modelis appropriate if the events are independent and if there is no aging, i.e., failure probabilities ($) are constant.
~b) The Poisson distribution is commonly used in PRA to model processes in which events are generated over time. This model is appropriate if the events are independent and if there is no aging, i.e., the hazard rate is constant over time: f A(t) = Ac.
c) If the above conditions are not true (e.g., aging is an important phenomenon over the time scale of inteitst), a different process model may be needed.
- 3) If the random variable is continuous, determine if the random variable is restricted to certain ranges (e.g., if the variable takes on only positive values).
a) If the random variable is positively valued (as is the case with failure times), determine if the events are independent and if there is aging. 14
i i) If the events are independent and if there is no aging, the exponential distribution is an appropriate model, ii) If the events are independent but there is aging, one of a number of candidate aging models (e.g., Weibull, linear aging [l'4]) may be selected. .l-iii) If the events are dependent, a problem-specific process model will be needed. b) If the random variable is generated by an additive process, that is, Z-2Xkk (17) a normal distribution model may be appropriate. A random walk (e.g., see [15]) is a classical example of such an additive process. c) If the random variable is generated by a multiplicative process, i.e., Z -]Xk k (18) a lognormal distribution may be appropriate. It can be seen that in a number of situations, problem-specific models may have to be developed. Examples are provided in the following section to illustrate how this can be accomplished. 4.2 Likelihood Function Construction - Empirical Data Once a model has been developed for the process generating the data, it is generally a straightforward procedure to construct the likelihood function for a given set of data. The likelihood function, MBO), is proportional to the pmbability of observing evidence E, a.;suming that the parameter 0 is actually known. In the case of multiple, conditionally independent sets of evidence E, the likelihood function is given by L(El 0)-{MEi l0) (19)
*Ibe likelihood function for the ith set of evidence can be developed using appropriate distribution functions, as shown by the following examples.
l i 15 l
- 1) Binomial distribution a) Evidence: E = {r failures in a tdals}
This is the standard form of evidence used in PRA applications for demand failures. The likelihood function is shown in Eq. (1). I b) Evidence: E = {S,S,F,F,S} Here, "S" stands for success and F" stands for failure; the evidence is more detailed than that for (a) in that we now know the precise order of successes and failures in the sample. The likelihood function is simply derived using the laws of probability: 2 L(EiI ()-(1- 4)-(1 -4) 4 $-(1-4)- 9 (i_4)3 (20) It is interesting to note that although Eq. (20) lacks the leading coefficient of Eq. (1), it leads to the same posterior distribution as does Eq. (1) (when r = 2 and n = 5). This can be seen to be a direct consequence of the underlying Bernoulli process model, which assumes that the actual ordedng of successes and failures is irrelevant. More generally, examination of Eq. (5) shows that proportional likelihood functions will lead to the same posterior distribution (assuming the same prior is maintained), since they differ only by a constant which appears in both the numerator and denominator of Bayes' Theorem.
- 2) Poisson distribution a) Evidence: E = {r failures in the time interval [0,t]}
This is the standard form of evidence used in PRA applications for operating failures. The likelihood function is given by Eq. (2). b) Evidencc 4 = {less than m failures in the time interval [0,t]} The probability that less than m failures have occurred, given t and A, is derived by summing contributions from the different possible values of r (the random number offailures):
.-ig)r -b L(EilA)- d (2l)
- 3) B&tial distribution The exponential distribution models a process in which the random variable (usually time to failure) is continuous and positive-valued, events are independent, and aging is not present.
16
This process is intimately associated with the Poisson process; if the number of events in a given time interval is Poisson-distributed, the event interarrival times are exponentially distributed (with independent, identical exponential distributions). As in the preceding i cases, the precise form of the likelihood function depends on the precise form of the , evidence. Three examples follow. - a) Evidence: 4 = {ti,t2,... .t3 ,... .t,} By definition of the probability density function, the likelihood of the jth observation is given by the exponential probability density function evaluated at time t:3 L(tjl A)- Ae-Ab (22) Per Eq. (19), the likelihood function for a set of independent observations is given + by the product of sirritar terms: l i L(Ei l A)- h,., Ae-A'J -A -tj A'exp. (23) b) Evidence: K = {a < t < b}
)
l This case is similar to case (2b) above. The likelihood function is detennined using the cumulative distribution function for the failure time: b L(Ei i A)-fAe-Mdt - e-la - e-Ab (24) j c) Evidence: E = (E3 ,E 2 ), E = {ti2 3
,t ,....t3 ,... t,}, E2= {a < t,.i < b}
This case combines cases (3a) and (3b). From Eq. (19), the likelihood function is just the product of Eqs. (23) and (24): L(E I A)- 14E 1 A) L(E2 l A) (25)
- A" e-O')(e-*' - e-*)
4.3 A More Complicated Example The binomial, Poisson, and exponential distributions discussed in Section 4.2 are directly useful for a wide variety of PRA estimation problems. However, situations can arise where more complicated likelihood functions need to be constructed. Given a process model, general approaches for developing functions of random variables (e.g., see [15]) can be used to develop likelihood functions. 17
As a simple illustration, consider a simple model for fire duration in which the total
. duration of the fire (T,) is the sum of the fire detection time (T,) and the fire suppression time (T,):
Tr - Td + T. (26) Assume that T, and T, are independent, exponentially distributed variables with characteristic parameters A, and A,, respectively. Further assume that data are available for T, but not for T, or T, individually. The probability of observing a particular time T is proportional to the probability density function for T, evaluated at T: AdAs .x ,, _ .A ,, , d-fTi(T I Ad,b) . (27) A 2 7,- At x, . x, . x Eq. (27) is obtained in the usual manner for the sum of two independent random variables, i.e., by convolving the distributions for T, and T,: s f Ti(T lld.b)-ffT,(t i A4)f T.(t- ti L)dt
, (28)
-[A4e- Ad'k e-As(5-8) dt 0
Using Eqs. (19) and (27), the likelihood function for a string of independent observations for T,can be easily constructed: L(ti,t 2,...,ti...., t. I Aa,M - fr,(ti I L ,L) (29) y. It is interesting to observe that although the aleatory variables T, and T, are conditionally independent (by assumption), the parameters A, and A, are probabilistically dependent. In other words, the joint posterior distribution for A, and A, is not equal to the product of the marginal posterior distributions: xj(Ad,klE)* xj(AalE) ni(AIE) (30) The dependency arises because the evidence does not discriminate between its detection and suppression constituents. The examples in Sections 4.2 and 4.3 show that, in the case of empirical data, the role of the likelihood function in the estimation process often is to incorporate the data within a model that describes the aleatory behavior of the system being studied. .An example showing the use of a different form of evidence is discussed next. e 18 \ .
- t l
4.4 Treatment of Expert Opinion Expert judgment, from the standpoint of Bayesian estimation, simply represents a different form of evidence. Thus, as long as an appropriate likelihood function can be generated, expert ..- judgment can be used in the estimation process. In this section, we follow the approach used by Ref.16 to treat a simple problem. More complicated cases have been investigated and are l discussed in the literature (e.g., see [17,18]). Let us assume that we ask a single expert to provide a point estimate for the probability of failure on demand ($) for our standby pump.' Denote the expen's estimate by $*. How do we treat this piece of evidence? To answer this question, recall that the likelihood function is the pmbability of observing l (receiving) the evidence $*, assuming that the tme value of(is known. Because probability is an l internal measure of likelihood, we emphasize that this function provides the analyst's pmbability of observing $*. In other words, it provides the analyst's assessment of the expen's expenise. For instance,if the expert is believed to be perfect, the likelihood function is a Dirac-6 function located at$=$*: L($* I $)- 6($ *-$) (31) In other words, the analyst is completely certain that the expert's estimate will be equal to the true value. On the other hand, if the analyst has no confidence in the expert, the likelihood function is flat: L($* I $)- C (32) where C is a constant. Regardless of the tme value of $, the expert is expected to provide an estimate (essentially at random) from anywhere in the admissible range, O s 4 s 1. In more realistic situations, the analyst has partial faith in the expen. The lognormal distribution has been proposed as a simple model for these situations: 1 1 fin &*-In(B$) 2-L($* I() 42no$
- exp (33) 26 o .
where B is a bias factor and o is a dispersion factor. Note that, according to Eq. (33), the median value of $* is B4. The bias factor therefore measures the analyst's assessment of the tendency of the expen to underestimate (B < 1) or overestimate (B > 1) the tme value of $. 8 Note that the issue dealt with in this paper is the use of expert opinion in the Bayesian estimation process; the issue of expert opinion e'icitation is a separate. extremely important, topic. 19
i - The dispersion factor is easiest to interpret if there is no bias (i.e., B = 1); in this case, o is a direct measure of the expen's expenise. A small value of a implies that the expen is likely to produce an estimate close to the true value. (In the limit as o approaches 0, the lognormal likelihood function approaches a 6-function about $.) A large value of a implies that the expert may provide a good estimate, but is quite likely to provide a bad one. When the expert is biased, o ./* still measures the dispersion of the likelihood about the median. Now, a small value of a implies that, in the judgment of the analyst, the expert is likely to provide an estunate close to the true value moddied for bias (B$). It is important to observe that in this formalism, both B and a must be provided by the analyst. Although these may be measurable in principle, they are difficult to assess for the data-less, controversial situations in which expert opinion is typically called upon. Perhaps for this reason, many PRAs to date have not employed a formal Bayesian approach for incorporating expert opinion;instead, they have employed such methods as arithmetic or geometric averaging of the experts' estimates (e.g., see [19]). However, if expert opinions are to be combined with data, then the Bayesian formalism is needed. 20 1
- 5. PRIOR DISTRIBUTIONS Development of an appropriate prior distribution is, sometimes with just cause, the most controversial part of a Bayesian analysis. It can often be the most resource-intensive, as well.
This section discusses some of the issues to be considered when creating an informative prior .l-distribution (i.e., a prior distribution that at least partially reflects the analyst's state of knowledge) as well as a number of so-called " objective," noninformative prior distributions. 5.1 Informative Prior Distributions An informative prior distribution is one that reflects the analyst's beliefs concerning an unknown parameter. (This is in contrast to mathematically defined noninformative prior distributions discussed laterin this section.) The development of an informative prior distribution can be, in principle, a challenging process as it requires the analyst to convert his/her own intemal (and typically qualitative) notions of likelihood into quantitative measures. In practice, however, the process is often less difficult for two reasons. First, data are often available for the updating process. This means that the prior , distribution does not have to be specified to an extraordinary degree of accuracy, since the effect of the prior distribution diminishes with increasing amounts of data (see Section 3). Often, a rough representation of the analyst's state of knowledge is sufficient. -(As will be shown later,
" noninformative" prior distributions can be viewed as a result of taking this argument to its logical extreme.) For this reason, such commonly used ad hoc procedures as the identification of " upper and lower bounds" and the fitting of a lognormal distribution to these " bounds' can yield results satisfactory for the purposes of decision support, despite their lack of rigor.
Second, the Bayesian updating process governed by Eq. (5) does not always start from scratch. Instead, the process is iterative; the prior distribution represents the analyst's state of 1 knowledge prior to the latest set of evidence, not prior to all evidence. [Eq. (5) could emphasize this point by using notation such as gx for the prior and x, for the posterior. However, this would not match current PRA conventioia ' lor labeling the prior and posterior.] In many cases, therefore, the informative prior distribution is simply the posterior distribution from the last updating calculation. In cases where updating has not yet been performed, a variety of formal techniques are 3 available to incorposte available information into the prior. Three such approaches that have been j used in PRA, the two-stage Bayes approach, the empirical Bayes approach, and the method of ' maximum entropy, are discussed below. Depending on the particular problem at hand, these approaches can be somewhat complicated to execute. This complication can be viewed as the price to be paid for a more structured and perhaps more robust prior distribution. 5.1.1 Two-Stage Baves Method One of the potential problems with conventional applications of Bayes Theorem is that, although data are typically collected from a variety of sources (e.g., different plants with diffe ent underlying parameters ofinterest), they are treated as coming from a single source. The result is 21
that uncertainties due to source-to-source variability are ignored, and the resulting posterior distribution tends to be too narrow. To show this, consider our cadier problem of estimating a standby pump's demand failure probability,4. Evidence (data) from a variety of plants may be available of the form .' E - {(ri,ni),(r2,n2),....(ri,ni),..,(rm,nm)} (34) where r, is the number of failures and n, is the number of demands (trials) at plant i. A straightforward analysis would employ the sums of the failures and demands, a m r- ri and n - ni directly in the binomial likelihood function of Eq. (1). However, this approach does not recognize that there may be significant differences among plants (e.g., different maintenance practices), leading to a different failure probability at each plant. Thus, although at fiist glance the use of the summed (or " lumped") data tends to yield " good results" (a relatively narrow posterior distribution), it must be recognized that these results reflect a population-averaged $. They may not adequately reflect the full range of uncertainty appropriate for a particular plant's failure probability. The two-stage Bayes method, first described in a PRA context by Kaplan [20], is a modeling approach designed to address plant-to-plant (or, more generally, source-to-source) variability explicitly. Deely and Undley [21] provide additional useful discussion on the approach. The general procedure in the two-stage Bayes method is to create a prior distribution for 4 based on population data (the first stage), and then to update this prior distribution using plant-specific data (the second stage). Tais results in a plant-specific posterior distribution for 4. The second stage involves a straightforward application of Bayes' Theorem, using the ideas discussed cadier in Section 3. The first stage requires additional modeling to represent plant-to-plant variability and is the focus of the following discussion. l To begin, let us assume that we know (with perfect certainty) the values of the plant-specific demand failure probabilities for all plants, except for the one we are currently analyzing. An illustrative, hypothetical plot of these failure probabilities for 8 plants is shown in Figure 4. m - so ,e N r-E E1 i E E i di i i i l ll l ll l l >$ e m E % i i Figure 4 -Illustrative Plot of Plant-Specific Failure Probabilities 1 22
- 3 it can be seen that Figure 4 could be used to generate a histogram for a random variable 4, and that,if there were a sufficiently large number of plants, this histogram could be approximated r.asonably with a smooth probability density function. This density function, shown in Figure 5, is called the population variability curve (PVC); we denote it by g($lg), where 2 is the vector of parameters characterizing the curve. *
\
G 4 4 s Figure 5 - Hypot!".tical Population Variability Curve l Recall that we wish to estimate the demand failure probability for a particular plant (say plant k). The PVCis useful because if the only thing we know about plant k is that it belongs to the population represented in Figure 5 (that is, if we have no reason to believe that plant k is necessarily much better or much worse than the rest of the plants), then the PVC provides a , reasonable priordistribution for $,. (This is the key premise of the two-stage Bayes approach; if 1 we believe that 4, is likely to appear'far out in the tails of the PVC, the following procedure for developing an infonnative prior distribution does riet apply.) In reality, we do not know the exact form of the PVC. In other words, the parsneter vector 1ic uncertain. The first stage of the two-stage Bayes approach therefore is conc;med with { developing a posterior distribution for1. This posterior distribution will then be used to generate a j population-averaged PVC. i l To make the discussion more concrete, we will assume that the PVC is a beta distribution. [Lognormal distributions can be used when estimating operating failure rates or very small demand i failure probabilities (small in the sense that any tail truncation above $ = 1 has a negligible effect).] Thus,! = (a,p), where l l g($Ia,p) r(a)r(p)$"~1(1-$)bI (35) 1 The posterior distribution of the vector (a,$) is calculated using the two4imensional form of Bayes' Theorem-4 1
t *
*0 Ki(% I E) ,, "' '
(36) ffL(EIa, )so(a,p) dad $ 00 where the evidence Eis as indicated in Eq. (34). In the absence of any information concerning a and p, a uniform prior distribution for (a, )is ofter, used. (P6rn [22] constructs non-infonnative prior distributions for a similar problem; the general topic of non-irformative priors is discussed laterin this section.) The likelihood function is derived from tl% nodeling assumptions: a) the demand failure rates $3 charactetize a Bernoulli process at each plant, b) the PVC is beta, and c) the data sets from each plant are conflitionally independent. Considering only the evidence from plant i, the appropriate likelihood function is the so-called " beta-binomial": L(EiI a, )- (ri Ini,$,)g($ Ia,p)d$, 4
" d$i (37)
-fo ri! (ni - ri) 4(1-43)"'~ r(u)r(s)$f-3(1-$,)b' r(ni +1) ,
r(a 8) , r(ri + a)r(ni-r, + p) r(ri+1)r(ni-ri+1) r(a)T( ) f(ni + a + p) where the first term in the top-most integrand represents the conditional probability of observing r, failures in n, trials, given 4,, and the second term is the conditional probability that the failure rate is
$,, given a and p. The likelihood function for the entire set of evidence is the product of the likelihood frinctions for the individual plants (excluding the plant being analyzed):
L(E<a,p)- ]L(EiIg) (38) i:t With the posterior distribi tion for (a,5), i.e., x (a,$lE), 3 the expected (average) PVC < a be developed: g($ 1 E)-ffg($1 gp)ni(a, l E) dad UU (39)
- $"-l(1-$[~'ai(wpl E)dadp Following the previously presented arguments, g($lE) can now be used as an informative prior distribution,i.e., n ($),
o for the second stage of the two-stage Bayes analysis, during which 24
[ . , . ,: ,
- e plant-specific data are incorporated using the conventional Bayesian methods described in Section 3 of this paper.
The computations required by a two-stage Bayes analysis, while not trivial, can be performed using either dedicated computer programs, or generic equation-solving packages. .- Extensions of the method to treat multiparameter estimation for PRA have addressed particular cases (e.g., [22,23]); general extensions can be addressed using hierarchical Bayes methods discussed below. While there are many situations where the sophistication of the approach is not needed,it should be pointed out that there are also a number of examples where the approach l results in distributions that are much (e.g., orders of magnitude) broader than those obtained using a conventional," lumped data" approach (e.g., [24]). In these cases, the two-stage Bayes method weakens the impact of generic data on the final results. The two-stage Bayes approach is really a particular case of a more general approach, referred to in the statistical community as hierarchical Bayes. The terminology used by the statistical community (e.g., regarding " stages")is somewhat at odds with that used by Kaplan, as discussed below. Good discussions on hierarchical Bayes analysis can be found in Refs. 25-27. In a hierarchical Bayes approach, the prior distribution is developed in multiple stages. In i the example used above, the first stage prior would be g($la,$). The parameters that characterize l this prior, a and , are referred to as "hyperparameters." As in the two-stage Bayes approach, the hierarchical Bayes approach allows these hyperparameters to be uncertain; this uncertainty is expressed using a second-stage prior distribution. Unlike the two-stage Bayes approach, the parameters of this second-stage distribution can also be uncertain. In such a case, the uncettainty I in the second stage parameters would be characterized using a third-stage distribution. This modeling process continues until, at some point, a stage is reached where the prior distribution I parameters are assumed to be constant. This final prior distribution is called the "hyperprior." Comparing the hierarchical Bayes approach with the two-stage Bayes approach, it can be seen that the first stage of the latter encompasses both stages one and two of the former. (In the hierarchical Bayes approach, the average PVC is called a hierarchical prior.) Similar to the two-l stage Bayes approach, efforts to develop an informative hyperprior are usually limited; non-informative distributions are generally used. Unlike the two-stage Bayes approach, the hierarchical Bayes approach does not require that the first stage prior represent population variability; it simply ! acknowledges that the parameters of g(*) may be uncertain. As another difference, when the hierarchical Bayes first stage prior is interpreted as representing population variability, it typically I does not exclude data from the plant of interest [i.e., plant k in Eq. (38)], as is advocated by , l Kaplan [20]. However, this lauer difference is not expected to lead to significantly different l l bottom-line results for most practical cases of interest. , I l 5.1.2 EmniricalBaves A somewhat simpler method for generating an informative prior distribution to account for population variability is the " empirical Bayes" method [26-28]. This method involves the use of 1 classical techniques to fit the prior distribution to available data. As will be seen, it is not purely 25 I
1 Bayesian in philosophy. Nevertheless, it is a pragmatic approach to creating a prior distribution when data are available. Empirical Bayes approaches can be classified generally as either parametric (PEB) or nonparametric (NPEB), depending on whether the prior is assumed to have some specified ' functional form. The parametric approach has been the one taken most often in PRA-ielated applications (e.g., [29,30]). However, the nonparametric approach has seen some recent application. To illustrate the PEB approach in its simplest form, consider the earlier problem of using generic data discussed in connection with the two-stage Bayes method. As in the two-stage Bayes method, we assume that the prior distribution for the probability of failure 4 (for a specific plant) is reasonably modeled by a population variability curve (which treats the plant-to-plant variability in (). We further assume that the population variability curve is beta [see Eq. (35)]. Using available data, with each observation assumed to be conditionally independent [see Eq. (34)], we wish to estimate the parameter vector (a, ). In the two-stage Bayes method, the analyst develops a posterior distribution for (a,p), x,(n, IE); the prior distribution for 4, g($lE), is then obtained by averaging the population variability curve over x (a,plE).' In the PEB method, on the other hand, 3 the analyst develops point estimates for a and . These estimates, together with Eq. (35), define the pnor distribution. Note that the PEB approach does not account for epistemic uncertainties in aandp. As in most classical estimation problems, a number of different point estimates can be developed for a and . We consider two representative estimation methods discussed in by Shuitis, et al [31]; Berger [26] and Martz and Waller [32] provide additional discussion on the empirical Bayes approach in connection with the problem of treating different types of generic data (e.g., parameter estimates as well as raw data). f I
- Moment-Matching Methods One simple classical method to determine a and p is to match the prior distribution's moments with the data moments. [Shultis et al. call this the Prior Moment Matching Method (PMMM).] Since the prior distribution is assumed to be beta, this means that a and can be found by solving the following equations:
4 a+p op 1
" - (40)
(a+)2[y,p,3) , _3 },q (h-$)2 where T" $i
$ - m1",.3 8
Note that, in general, the average population variability curve obtamed using the two-stage Bayes method is not beta (or lognormal) in shape. In fact,it may not be even remotely beta or lognormal. 26 L
=
.M o t
and where the coefficient 1/(m-1)is used to develop an unbiased estimator of the variance. Since the (, are not actually known, they are estimated using the available data:
$, - '!. (41) ,,
ni - A second moments-based approach which is theoretically more appropriate and more widely used than the PMMM approach uses the moments of the marginal distribution for the observed random variable, the number of failures [see Eq. (37)]. Hence, this approach is sometimes referred to as the marginal moment-matching methods, or MMMM. It recognizes that if population variability is described by the selected prior distribution, then the observed failures will constitute a random sample from the marginal distribution. The mean and variance of the marginal distribution are then used to find estimates of a and p. This method is illustrated with an example in Section 6. Maximum Ukelihood Method A second classical technique for determining point estimates for a and $ is the method of maximum likelihood. [Shultis et al. refer to this approach as the Marginal Maximum Ukelihood Method (MMLM), because it uses the marginal distribution to develop information about the prior. in the context of empirical Bayes analysis, the statistical community often refers to it as the MIeII approach.] The likelihood function to be maximized is the same as that in the two-stage Bayes approach. Thus, for the ith plant, the unconditional (" marginal") probability of observing r, events in n, demands is obtained by averaging the binomial distribution over the prior distribution for 4,, as shown by Eq. (37). The likelihood of observing the total evidence set Eis then given by: L(E I a, )- L(ElIa,p) (42) i=1 Eq. (42) reflects the fact that in most PEB analyses (both moment matching and ML-II), as well as in most hierarchical Bayes' analyses, data from the ith plant is not excluded when developing the prior. This differs from the approach described in Kaplan [see Eq. (38)], but is not expected to have a significant impact in most practical applications. The maximum likelihood estimators for a and p are obtained by maximizing the likelihood function (or, equivalently, the log-like'ihood function). Both of the PEB methods described use data in a relatively straightforward manner. With respect to its application to the treatment of data exhibiting population variability, however, it should be noted that the resulting prior distributions will often be narrower than one produced from the two-stage or hierarchical Bayes methods. This is because the PEB approxhes do not incorporate epistemic uncertainties in the PVC parameters (e.g., a and p) into the final result. There are techniques that can be used to incorporate variability in the parameter estimates into the ( results, giving a better approximation to a fully Bayesian analysis. One technique makes use of the asymptotic properties of the maximum likelihood estimators and, hence, can only be used in the 27
MI.eII approach. Kass and Steffey [33] provides additional details, while Martz, Kvam and Atwood [34] illustrate this more accurate PEB method for a typical PRA application. Note that none of the PEB approaches is guaranteed to give reasonable parameter estimates. The MIIs can fail to exist, or there may be convergence difficulties to overcome when maximizing .. a complicated likelihood function. As for the moment estimates, they can tum out to be negative, even though the model requires them to be strictly positive. In contrast to the PEB approach, the nonparametric Empirical Bayes (NPEB) approach does not assume a specific functional form for the prior distribution. Instead, it uses past data to bypass having to specify a prior. NPEB has not seen wide use in PRA and will not be discussed further bere. The interested reader is referred to Refs. 26, 27 and 32 for introductions to the eyyru.ch and references to the literature. An example of its application to the issue of emergency diesel generator unavailability is provided by Samanta, et al. [35]. 5.I.3 Maximum Entmov Prior Distnbutions The previous methods for constructing informative priors are based en the use of empirical data. In situations where such data are not available, an informative prior distribution can be constmeted if other forms of information are available. This section discusses a method for constructing a prior distribution based on a given number of specified constraints. In this method, one develops a prior distribution which is as " noninformative" or vague as possible aside from the specified constraints. (Typical constraints include a specified unge for the piu.ne ofinterest, e.g., O s $ s I, specified distribution moments, or specified distribution percentiles.) To construct such a distribution, some measure of the information content of a distribution is needed. One such measure is the " entropy' of the distribution, developed by Shannon in the 1940s. Before using this measure to develop prior distributions, we provide a briefintroduction to the notion of entropy. A very readable intmduction to the subject can be found in Br6maud [36]. Consider a set of observations {xi , x2. - . X.}, and denote the probability of the ith observation by x(x,). We define a statistic called "information" as follows: 1(xi) = -In[x(xi)] (43) (Many definitions of "information" and " entropy" employ the common logarithm; our use of the natumi logarithm does not chang'e the basic approach and is somewhat more convenient.) Although this mathematically defined statistic does not necessarily correspond to any philosophical nations of information (for example, it does not consider the value of x, in determining the value of information), it has two useful properties. (i) 1(x,) < 1(x3 )iff x(x,) > x(x)). In other words, less likely observations convey more information. 28 l
- s (ii) 1(x,,xj ) = 1(x,) + 1(x)).
Thus, the information content of a joint observation equals the sum of the infonnation for ! the separate observations.
- e. !
The entropy for the sample {xi, x2 , ... , x,} (sometimes refened to as the "unceitainty") is defined as the average amount ofinformation: (44) H(1)- s--}1x(xi)tn[n(xi)] , j Although this definition of entropy is given with respect to a sample, it applies to discrete probability distributions as well. For example, consider the one-point distribution p(x,) = 1 for i =j, p(x ) i= 0 for i
- j. Applying Eq. (44), the entropy of this distribution is 0. This result is !
- consistent with an intuitive notion of entropy, as there is no uncertainty in this distribution. At the other extreme of suformation content, consider the distribution p(x,) = 1/n for i = 1, 2, ... , n. This I distribution assigns equal probability to each point. It can be shown that the entropy of this unifonn distribution, inn, is the largest possible entropy of all possible discrete distributions for x (see [36]). Once more, the entropy definition of Eq. (44)is consistent with intuition.
In general, it can be seen that, given an arbitrary distribution, a unique entropy value (H) can le computed. Our problem is somewhat different: we would like to find the distribution that maxunizes H, subject to a limited number of constraints. Noting that in PRA, one is more often interested in continuous rather than discrete prior distributions, we first need to extend the entropy definition provided in Eq. (44). An obvious extension is as follows, where 0 is the continuous parameter ofinterest: H = -fn(0)ln[x(0)]de (45) Using the calculus of variations, it can be shown that when constraints on distribution moments or percentiles are specified, a prior distribution form that maximizes H can often be l derived [26]. A number of constraints and resulting maximum entropy prior distributions useful for PRA applications are provided in Table 2. 29
Table 2 - Some Maximum Entropy Prior Distributions Potentially Useful for PRA Constraint (s) Prior Distribution 1 '* as8sb y_, O a:0 .I. earp 40] = known P Se* e* - c" d asesb 40] = p known where $ ( w 0) satisfies be* - ae" 1 E* epb_ eps ~f
-= < 0 < =
I I_ ' 0 E0] = known q 2 _2 o Var [0] = a known , it should be recognized that there are two technical problems with the approach described above. First, the simple definition of entropy for continuous distributions provided in Eq. (45) has three potential drawbacks: a) H may be negative, b) H is not necessarily bounded, and c) H is not invanant under coordinate transformations. An alternative definition of entropy is: n(0) x (0)de (46) H = -f,,in ,xm(0), where x,,(0)is the " natural" noninformative distribution for 9, i.e., the distribution which tends to maximize H in the absence of any constraints. This definition, which leads to what have come to be called ' constrained noninformative prior distributins," will tend to reduce the importance of the third problem (the transformation noninvariance of d). Refs. 26, 37 and 38 can be consulted for more information on this alternative definition of entropy and the results it produces. It should be cautioned that this definition is not without problems. For example, the choice of a " natural" noninformative prior distribution is far from clear. As will be discussed in Section 5.2, the form of a noninformative prior depends on the particular definition of noninformativeness employed; this introduces ambiguity into the dermition of H. 30
Second, there are some situations where a maximum entropy prior cannot be derived. (An example is when 6 can take on any real value and only the mean value of 0 is specified.) However, the biggest problem may be not with how the maximum entropy distributions are " derived but rather how they are used. In this discussion, we have presented the method of maximum entropy as a reasonable approach for generating a prior distribution based on limited I prior information. Indeed a number of fairly recent PRAs (e.g., [39,40]) have employed maximum entropy distributions. However, care should be taken to avoid using the maximum entropy approach as a means to avoid Bayesian updating. Specifically, the analyst should not use new data to directly generate a new maximum entropy distribution. Although such an approach is a pragmatic means for dealing with uncertainty, it has some philosophical as well as practical shortcomings. l Philosophically, there are three objections to this approach. First, the maximum entmpy distributions used in PRAs are intended to deal with epistemic uncertainties, and thus fall within l the framework of the subjective interpretation of probability. We have seen earlier that, if the , l analyst's probabilities are to be updated according to the laws of probability, Bayes' Theorem is ! l the only updating mechanism that can be used. Thus, avoiding the use of Bayes' Theorem can result in incoherent probability assessments. Second, although the interpretation of entropy is fairly clear in the case of discrete variables, it is quite less clear in the case of continuous variables. The maximization of H is a formal, mathematical process, but the lack of an intuitively meaningful, unique interpretation (or even definition) of H means that there are no overriding reasons to maximize H as opposed to other functionals of x(0). Third, the constraints employed in the maximization process are supposed to be factual, that is, subject to no uncertainty. In practical PRA applications, efforts are rarely taken to ensure that the constraints (which are on the characteristics of the epistemic distribution) are specified to such a degree of certainty. Thus, a degree of subjectivity is introduced into the analysis, yet this subjectivity is rarely acknowledged. It can be seen that in practical PRA applications, the method of maximum entropy is not necessarily a "more objective" approach than the Bayesian approaches discussed earlier. Subjectivity is generally intmduced via the assumed constraints and through the basic decision that entropy, however it is defined, is the quantity to be maximized. We believe the method can be used to support the development of prior distributions, but recommend that it not be used as a l means to entirely avoid the Bayesian updating process; such an approach updates the analyst's conditional probabilities in a manner inconsistent with the laws of probability. 5.1.4 Cautions in Developine Informative Prior Distributions The precedmg discussions show that, when data (even data that come from sources not completely identical to the one being analyzed) or other information (e.g., constraints on the prior distribution) are available, there are a number of ways to formally and systematically constmet prior distributions. When such information is not available, the task can be moie difficult. (Again, one should keep in mind that the importance of precisely defining a prior distribution decreases with increasing amounts of data to update that prior. Also note that the decision which is to be supported by the estimation process may not be especially sensitive to the prior distribution.) l 31 1
There are a number of approaches for developing a prior distribution directly from the analyst's state of knowledge. In the case of discrete variables, lottery methods (in which the analyst develops a lottery equivalent to the problem at hand) can be employed. In the case of ,, continuous variables, the analyst can try to elicit his/her own beliefs concerning such key " characteristics as the distribution moments or percentiles (or, equivalently, its fractiles). Winkler and Hays [10] and Berger [26] provide additional useful discussion on the subject; Goel [41] identifies some software packages aimed at assisting the process. However, there currently is no agreed upon "best" method. Rather than focusing on the merits or demerits of any particular approach, the following discussion simply raises five warnings (presented in no particular order) that should be considered when developing an informative prior distribution.
- Beware of zero values.
Eq. (5) shows that if the prior density function for 0 is assigned a zero value for any range of 6, the posterior density function wi_Il be zero in that range, regardless of the data. In other words, the prior distribution will be impossible to update over that range. Clearly, zero values should not be assigned unless it is absolutely impossible for 6 to have a value in a given range (e.g., as is the case for negative failure rates). This issue is not generally a concern when standard functional forms (e.g., lognormal, gamma) are used to model the prior distribution. It is most likely to arise when using discrete probability distributions, histograms, or other distributions with finite bounds.
- Be aware of cognitive biases.
It is well known that experts and lay people alike tend to be overconfident in their quantification of uncertainty. This results in overty narrow distributions. The distributions can also reflect systematic biases introduced by the heuristics typically used when developing a prior distribution. Some of these heuristics are as f'ollows [11]: Availability. The assessor has personal experience with the event, which tends to increase his/her own assessment of the likelihood of the event. Conversely, if the assessor has no personal experience with the event, his/her probability may be given toolow a value. Anchoring and Adjustment. The assessor fixes a value (anchor) for an event with which he/she is familiar, and then adjusts the remainder of the distribution about this anchor. The problem is that the assessor may be too confident in the anchor value selected (or may become too confident as the analysis proceeds), and may not allow for sufficiently large deviations about the anchor. Representativeness. The assessed probability is based on the degree to which the event mirrors the assessor's internal notion of the undedying process. As a simple 32
4
. .\ . ,
i example,the series of coin toss results {H,H,T,H,T} is often thought to be more likely than {H,H,H,T,T} because it appears "more random." i These biases are not necessarily avoidable. However, with awareness of their sources, an analyst can attempt to compensate for them at least to some degree. .- Beware of generating overly narrow prior distributions. The previous bullet concems the generation of overly narrow prior distributions due to cognitive biases. Overly narrow prior distributions can also arise in other ways. A typical example concems situations where generic industry data are used to generate a prior distribution for a specific plant. If a great deal of generic data are available, and if plant-to-plant variability is present but not recognized, an extremely and unreasonably peaked prior distribution can result. In some recent cases, it has been observed that the generic prior distributions are so narrow that they overwhelm the effect of even fairly large amounts of data. The reason for this pmblem is that a generic prior distribution that does not account for plant-to-plant variability effectively models the population average behavior. With large amounts of generic data, we tend to know this average behavior very well, and such a lumped prior distribution will indeed be quite narrow. However, this generic prior i distribution is not necessarily the prior distribution that should be used for a plant-specific analysis (since we have no assurance that the plant being analyzed should behave as the l population average). It can be seen that if even large amounts of data have little or no effect on the posterior distribution, the prior distribution needs to be examined carefully, especially in the case of generic prior distributions. Ensure that the evidence used to generate the prior distribution is relevant to the estimation problem. Obvious as this may sound, there are practical situations where this can be a concern. Ref. 13 observes that evidence may be collected under normal operating conditions, whereas the PRA model is addressing system response under accident conditions. Note that issues associated with the applicability of evidence used in the updating of the prior distribution, as opposed to the development of the prior distribution itself, are discussed in Section 7. If there is a concem with the applicability of a particular body of evidence, the analyst should consider generating a prior distribution that excludes this evidence (using the methods of this section) and then updating the distribution using the methods discussed in Section 7. l 33 ) 1 I I
7
- Be careful when assessing parameters that are not directly observable.
The prior distribution is supposed to reflect the analyst's beliefs conceming a particular parameter. If the analyst has little direct experience with the parameter, it can be difficult to justify an informative prior distribution. .."
)
)
For example, as part of the two-stage and hierarchical Bayes approaches described earlier, ! the analyst must develop a hyperprior for a .and $. Since these parameters are mathematical entities and are not directly observable, it is difficult to envision how probabilities can be dimetly assigned to different (a,$) regions. Indeed, a common practice in this case is to relate a and p to the distribution characteristics (e.g., the median and error factor), which are somewhat more intuitively meaningful quantities. The analyst's judgments on possible values of these characteristics can then be translated into judgments on a and p. Along these lines, it should be noted that the mean value of a distribution is also a mathematically defined quantity which may have little physical meaning, especially in the case of highly skewed distributions. (For example, one can easily generate sopormal distributions where the mean value is larger than the 95th or even the 99th percentile.) Thus, directjudgments on the mean must be considered carefully. We observe that one can take this point to its logical conclusion, and question the accuracy of direct judgments on any inferable but not directly observable characteristic parameter (e.g., the failure rate A). One alternative, as pointed out by Clarotti and Runggaldier [42], is to makejudgments on the likelihood of different values of an observable random variable (e.g., the failure time); these judgments can be used to infer a probability distribution for the characteristic parameter. An extension is to make judgments on the likelihood of hypothetical sets of values (i.e., samples) of the observable variable. In preliminary trials, we have found that this latter approach can lead to inferred distributions that do not necessarily correspond with those obtained by directjudgment.
- Beware of conservatism.
The natural tendency of an analyst when faced with uncertainty is to employ a conservative prior distribution. The problem is that the degree of conservatism can vary from analyst to analyst, thereby upsetting the ranking of risk contributors. More generally, a conservative analyst injects his/her own values into the analysis, and, to some extent, usurps the decision maker's role.
- Be careful when using discrete probability distributions.
One issue associated with the use of discrete probability distributions is related to the first caution concerning zero values. Discrete probability distributions (DPD), by their nature, effectively assign zero values outside the range of the DPD. This leads to trouble when data used in updating imply distribution shifts beyond the original range, as illustrated in 34
;.~.. \
Ref.13. (It also leads to losses of distribution variance when propagating uncertainties, but that is not an estimation issue.) A second, more subtle issue is that a DPD carries information in two ways: by selection of the DPD grid, and by the probability weights assigned to each grid point. Thus, for . example, even if equal probability weights are assigned to the grid points, the DPD does not represent a uniform distribution unless the grid points are evenly spaced. The analyst needs to take special care that the DPD does accurately represent his/her state of knowledge. DPDs are useful for dealing with inherently discrete parameters (e.g., weather categories) and for conceptualization purposes.- However, they can give rise to some difficulties when used as approximations for continuous distributions, as stated above. Curant modem computer hardware and software allow better numerical treatment of inherently continuous variables, and greatly reduce the need for DPD approximations. 5.2 Noninformative Prior Distributions l When large amounts of data are available for updating a prior distribution, and when the l analyst's prior beliefs are relatively vague, the effect of the data on the posterior distribution will tend to swamp any effect from the prior. It therefom is not very useful to spend large amounts of time and energy in carefully developing an informative prior distribution. In such situations, so-l called " noninformative" prior distributions, prior distributions which are mathematically I constructed to represent a vague state of knowledge, can be quite useful. Conversely, it must be cautioned that when there are litde data, the use of such noninformative prior distributions is more difficult tojustify. If the estimation problem involves a , significant contributor to risk, the analyst should try to develop an informative prior distribution, as this is likely to have a majorimpact on the results. As a practical matter, since PRA analyses tend to be iterative, one might start with a noninformative prior distribution (which, with its heavy tails, tends to be conservative). Informative prior distributions can then be developed for parameters which contribute significantly to risk. (Care must be taken in this iterative procedure to develop a prior that is as independent as possible of the data used in updating.) This section presents some reasons why a flat (uniform) prior distribution, the most intuitive noninformative distribution, is not necessarily the best, and discusses a number of different systems of noninformative prior distributions that have been used in PRA applications. l 5.2.I The Uniform Distribution Paradox The uniform distribution indicates that all values of the unknown parameter are equally likely. In other words, the analyst has no reason to believe that one particular value is more likely to be correct than another. Thus, the uniform distribution appears to be the most natural l distribution to represent a situation where the analyst knows nothing (or very little) about a l i-4 (e.g., $). 1 35 l j
'l - ! Now, it is reasonable to expect that if an analyst knows nothing about $, then he should know nothing about arbitrary transformations of 4 For example, the distribution of the transformed variable 9 = In$ should also be uniform. The problem is that the distributions for 4 and 9 cannot both be uniform. Indeed, the calculus of probabilities shows that we must have x($)d4- x(9)d9 (47) which implies that x(9)- x($)N d9 (48)
=eV if x($)is uniform.
This paradox shows that a more formal definition of a vague state of knowledge is needed to define a mathematically consistent " noninformative" probability distribution. As might be l expected, different definitions lead to different noninformative distributions. Two different systems of noninformative prior distributions are outlined below. Berger [26] provides references to the literature for other techniques. 5.2.2 Coniugate Pair Noninformative Distnbutions l An implicit way to define a vague state of knowledge is to look for prior distributions that, when updated, don't affect the posterior distribution [10]. This d is most easily exercised when dealing with likelihood function and prior distributions that make up a conjugate pair. As mentioned in Section 3.2, a conjugate likelihood function / prior distribution pair, when employed in Bayes' Theorem, leads to a posterior distribution which has the same functional form as the prior distribution. Table 1 presents two conjugate pairs (the binomial / beta and the Poisson / gamma) which are useful for PRA applications. In the case of the binomial / beta pair, it can be seen that if a = 0 and = 0 in the prior distribution, the posterior distribution parameters are completely determined by the data (r and n). Thus, these choices for a and p define one type of noninformative prior distribution: 1 (49) xo($) = $(1- 4) (0 s 4 s 1) Note that this distribution is improper. the integral over the range [0,1] does not exist. Moreover, it has singularities at the endpoints of the range (i.e., the density function goes to infinity). Such behavior turns out to be fairly common with noninformative prior distributions, providing additionaljustification for avoiding their use unless there are large amounts of data. The approach described above can also be used to develop a noninformative prior distribution for a Poisson likelihood function. The result is: 36
1 xo(A)= A (A=0) (50) 1 Again, the distribution is improper and has a singularity (this time at A = 0). 5.2.3 Data-TrandateA Tikelihood Functions and Noninformative Prior Distributions A second approach towards the development of noninformative prior distributions, described extensively by Box and Tiao [43], relies upon two notions:
- i The notion of a uniform distribution to represent maximum uncertainty is intuitively !
appealing. The question is if there is a "best" transformation of $ for which the distribution should be flat. l Itis not necessary to define the notion of maximum uncertainty in an absolute sense; it is only important to define it with respect to the available evidence (whose impact is quantified by the likelihood function). These twc, notions give rise to the following definition of a noninformative prior distribution: A noninformative prior distribution for 4 is one that is uniform in the transformed variable t = h($), where h($) is the transformation in which the likelihood function is " data translated," that is, the location of the likelihood function is specified by the data, but the shape of the likelihood function remains constant as $ changes. In other words, 9 is a location parameter. Figure 6 illustrates the concept of a data-translated likelihood function. With different sets of data, i the likelihood function shifts along the $ axis. However, the shape of the likelihood function remains unchanged. In effect, the prior distribution acts essentially like a constant over the range where the likelihood function differs significantly from zero. It is important to note that the likelihood function is only data-translated in the $ variable; L(B4) is not data-translated unless t = $. L(E lhp) 14E2hp) l l K0(9) I 9 Figure 6 - Data-Translated Likelihood Function 37
7 ' s Figure 6 also shows the noninfonnative prior distribution. The fact that its definition is influenced by the likelihood function [through the choice of transformation h($)] reflects the notion that the prior is noninformative with respect to the likelihood. To illustrate the concept, consider a situation where the random variable is lognormally .. distributed with parameters and o, where is known and ois unknown. We wish to develop a non-informative prior distribution for o. Assume the data are of the form a = {x,,...,x,}; the likelihood function is then given by 1N"*'~ (51) L(1I p,o)- f,. 42xox exp. 2( o , Dropping the dependence of14.) on because it is assumed to be known, it can be shown that "8 L(3 I o) m dexpf . (52) o' l 20 , 2 where s .1 2 (fax,_ )2 n ,.3 2 This is not data-translated in a because the data enter through the parameter s , and changes in s2lead to changes in shape as well as location of the likelihood function. Now consider the transformation h(o) = Inc. It can be easily shown that, after multiplying the likelihood function by s', one obtains , L(II Ino) = expf-n(Ino-Ins)-f exp[-2(Ino-Ins)) . (53) Thus, the likelihood function is data-translated in this transformed variable. As a result, the noninformative prior distribution is uniform in Inc. xo(Ino)- C (54) where C is a constant. Applying the usual rules of transformation [see Eq. (48)], I I xo(o) =*- (55) l 1 , The main problem with using this approach to find noninformative prior distributions in l other situations is that the transformation h(4) that yields a data-translated likelihood can be found for only a few special situations. However, transformations that yield approximately data-translated likelihoods can be found using Jeffrey's Rule, as discussed in many texts (e.g., [25,43]). Table 3 lists some non-informative prior distributions useful in PRA applications. 38
Table 3 - Some Useful Non-Informative Prior Distributions System Likelihood Function Prior Distribution Conjugate Pair Binomial , n! 1 r(g_ )n-r l r!(n - r)! $(1 -4) Conjugate Pair Poisson (h)' *.u L r! A Data-Translated, legnormal, o known Jeffrey's Rule 1 1 'Inx-p. 2 < constant KeXP' J . g Data-Translated, Imgnormal, p known Jeffrey's Rule
.2' 1
exp
' 1 'Inx 1
o Jeffrey's Rule Binomial
"' I
$'(1-$)"~'
r!(n - r)! Al- $) Jeffrey's Rule Poisson (At)' I r! * [ Note that the last two lines in Table 3 are the noninformative prior distributions refened to in Ref. 44. Updating these distributions with r failures in n demands or r failures in time t will lead to the following mean values for $ and A, respectively: 39
2r+1 R$1 n+1r+f - 2n+ 2 (56) 2r + 1 M A] r t+f - 2t ' 5.2.4 Cautions in Usinn Noninformative Prior Distributions The results in Table 3 show that most of the noninformative prior distributions of interest in PRA have singularities at one or both endpoints; many of these distributions are also improper (i.e., the integral of the density function over the entire range of the variable does not exist). In cases where data are present, these characteristics do not cause any practical difficulties. (Note that,in the case of the noninformative priors based on data-translated likelihood functions, the priors are defined to be noninformative with respect to the likelihood function. Thus, they are only applicable in the region where the likelihood function differs significantly from zero.) On the other hand, in cases where the data are weak, these characteristics can certainly impact the results. For example, let us assume that a particular valve at a plant has been tested seven times, with no observed failures. If we use the Jeffreys noninformative prior given by the 5th line in Table 3, we obtain a posterior mean of 0.06, which might be an overly conservative value for a valve whose typical failure probability on demand is on the order of 0.001. In this case, the prior mean value is 0.5, and this has a strong effect on the updated result when the data are weak, as they are in this case. Clearly, as stated earlier in this section, non-infonnative priors should be used only as screening tools if the evidence available for updating is weak. I f 1 i 40
, c.. ,.,
f
- 6. A NUMERICAL EXAMPLE We now illustrate the concepts presented in the sections with a simplified example. .
Assume that our population of interest consists of 10 emergency diesel generators (EDGs), similar I l to those used by commercial nuclear power plants in the U. S. as sources of emergency power. .... Assume our task is to estimate 43, the probability that the diesel at our plant (designated EDG No.
- 1) will fail to start on demand. At our plant, there have been 140 successful demands (i.e., no failures). The data for all 10 plants (keep in mind that we are interested in EDG No.1) are shown l in Table 4. I l
We shall first use the data for EDG No.1 only to dedve: a) classical point and interval l cstimates, and b) a Bayesian posterior distribution based on a non-informative prior distribution. We next employ the population data to derive posterior distributions using: c) the MI<II parametric empirical Bayes (PEB) approach, d) the MMMM PEB approach, and e) a two-stage Bayes approach. In all examples, it is assumed that the EDG fr.ilures are generated by a Bernoulli process, and so the binomial distribution can be used for the likelihood function. The summary results for all cases are pusented in Table 5. Table 4-Data for Example Problem EDG Failures Successes Demands 1 0 140 140 2 0 130 130 3 0 130 130 4 1 129 130 5 2 98 100 6 3 182 185
, 7 3 172 175 l 8 4 163 167 9 5 146 151 10 10 140 150 TOTAL 28 1430 1458 a) Classical Estimators The classical maximum likelibood estimate (MLE) of 4, is given by
!1 (57)
I) on 41 l l
let ($1,$u) denote the 90% confidence interval for $3 This is an estimator which expresses the uncertainty in our estimate of the true, but unknown value of 4,.' It can be determined using I tabulated values of the cumulative F-distribution. ) (ri + 1)Fo.95(2 ri + 2,2ni- 2ri) ,... ni- r + (r + 1)F0.95(2ri + 2,2ni- 2ri) (58) U
$t - ri + (n: - ri + 1) Fo.95(2ni- 2ri + 2,2ri)
Again, there is no uncertainty in $, in the classical framework. There is variability only in the estimator for $3;this variability stems from the aleatory nature of the process that gives rise to l the observed random variable (the number of failures in a given number of demands). In contrast, ' the Bayesian approach treats $3 as an uncertain parameter whose epistemic probability distribution expresses our lack of knowledge about the parameter's value. 1 b) One-Stage Bayes Updating j For this example, we employ the Jeffrey's non-informative prior distribution given in the fifth line of Table 3. The likelihood function is obtained using Eq. (1) and the data in Table 4: L(0 failures in 140 trials 14) - (1-$)18 (59) The resulting posterior distribution is a reverse J-shaped beta distribution with a = 0.5 and p = 140.5. In calculations (a) and (b), the estimates for 4, are developed only from the data for EDG No.1. In the following calculations, the data from the other 9 EDGs are used. Note that the failure experience appears to vary substantially from one EDG to another. This can be tested within the framework of classical statistics by testing the null hypothesis that the cells in Table 4 are independent. (This is equivalent to testing whether the EDGs all have the same probability of failure on demand in the binomial failure model.) Using a simple contingency table based on the Pearson chi-square test, it can be shown that the p-value from this test is 0.000766, which is very small and leads us to reject the null hypothesis of constant $. If the null hypothesis had not been rejected, we could have simply pooled (or " lumped") the data from the 10 EDGs together, re.iecting 1 the null hypothesis means that we should model diesel-to-diesel variability. c) Parametric Empirical Bayes (PEB), Maximum likelihood Method (ML-D) Assume that the prior is a beta distribution with unknown parameters a and p (see Table 1). 'Ihe MI II method uses the beta-binomial marginal distribution of the number of failures, which is given by Eq. (37). Assuming that each observation is an independent sample from this marginal distribution, we arrive at the following likelihood function: 8 Roughly speaking, this interval estimator will indude the true value of $3 90% of the time. 42
- . * :e .
t L(E I a,p)- "' + I) - I(" + 0) F(ri + a)T(ni-ri + )' (60) F(ri+1)T(ni-ri+1) F(a)T( ) F(ni + a + ) To find the maximumlikelihood estimates of a and p, we find the values of a and p that ,,- maximize the natural logarithm of L. Using a quasi-Newton algorithm, the maximum likelihood estimates are: 6-1.209 h - 63.45 The resulting prior distribution is beta with parameters & and h. Updating this prior distribution with the data for EDG No.1, the posterior distribution is also beta. Its parameters are: a' = 1.209 p' = 203.45 d) Parametric Empirical Bayes (PEB), Marginal Moment-Matching Method (MMMM) Similar to the MI II method, the MMMM also uses the beta-binomial marginal distribution shown in Eq. (37). Unlike the MIeII method, the MMMM uses the distribution mean and variance; it sets these moments equal to the sample mean and variance. The resulting set of two equations in two unknowns is solved to find the desired parameter estimates. Maitz and Waller [32] pmvide the following equations for use in estimating a and $: R p* g 6* - (E ~*
*2 Nw - M * -(N - M) where M = Number of EDGs (10in this case)
M N ='G}' ni . R= ri s. 2 w=- I"r - N ni The estimators of a and p are then given by l 43 ,
i
,.O ... ,
1 a*- *6* (62)
*-(1- *)6*
These estimates are computationally simpler to find than the MleH estimates; the latter can ,, be very difficult to find in some cases because of difficulties encountered in attempting to maximize the log-likelihood function. On the other hand, it should be noted that a* and $* may not be positive in some cases. For this example, the following estimates are derived: a* = 0.9676 p* = 49.415 Updating the resulting prior distribution with the data for EDG No. I results in the following parameter values: n' = 0.%76 p' = 189.415 The characteristic values obtained using this method (see Table 5) are quite close to those obtained using the MI II method. In practice, while it is better to use the MIeH estimates, the MMMM estimates are very easy to calculate. At the least, they can be used as starting points (when they are positive!) for the quasi-Newton algorithm used to find the MIeH estimates. c) Two-Stage Bayes The equations provided in Section 5.1.1 are directly applicable to the current problem. Using the evidence listed in Table 4 (and excluding the data for EDG No.1), the likelihood function provided in Eqs. (37) and (38), and a non-informative prior distribution proportional to (a + p)"[25], the resultingjoint posterior distribution for a and $ is shown in Figure 7. Noting that the ordinate is plotted on a natural log scale, it can be seen that the distribution exhibits a fairly sharp ridge which does not drop away very quickly as a and p increase. Both the high correlation between a and p (the correlation coefficient is nearly 0.9) and the slow decay with increasing values of a and are commonly observed for two-stage Bayes' analyses. The analyst must exercise great care in selecting the effective integration bounds during numerical evaluation in order to ensure that the contributions from the tails of the (a,p) distribution are not neglected. The posterior distribution for EDG No.1 is obtained by updating the average PVC [Eq. (39)] with the data for EDG No.1. Table 5 summarizes the results of the different updates performed in this section. Figure 8 shows the posterior density functions derived from the single-stage Bayes analysis (case b), the MIeH PEB analysis (case c), and the two-stage Bayes analysis (case e). Figure 8 also shows the average PVC (the prior distribution for the second stage) from the two-stage Bayes analysis. Note that upon close inspection, this distribution appears to have features characteristic of both a reverse 44
8 J-shaped distribution and of a unimodal distribution. Since the average PVC is a weighted sum a whole range of different distributions, this behavior can be attributed to the change in shape of the beta distribution as a changes fmm values less than one to values greater than one. However, it is worth noting that complex behavioris often also observed with the results of two-stage Bayes' analyses based on lognormal PVCS. *
.. . {
Enmination of Table 5 shows that, for the given set of data, the three analyses that treat population variability (Cases c, d, and c) lead to mean values and key percentiles that are quite close (by PRA standards). The results of the single-stage Bayes analysis (Case b), which does not ' employ data from other plants,is swble to those of Cases c, d, and e because the population data are not very strong (see the two-stage prior in Figure 8). The left-shift of the Case b posterior distribution is due to the reverse-J shape of the non-informative prior distribution; the data for Plant 1 (0 failures in 140 demands) do not preclude the possibility of very small failure rates. To better illustrate the impact of treating plant-to-plant variability, the last line of Table 5 ("InmaeA Data") shows the results that are obtained when all of the EDGs are assumed to come from a homogeneous population. In this case, the single-stage appmach outlined in Case b is employed to treat the data shown in the ' TOTAL" line of Table 4. The effective value of a is 28.5; the effective value of p is 1430.5. Contrasting the resulting density function in Figure 8 l with the density functions resulting from the other approaches, it can be seen that not treating population vanability results in a much narrower, right-shifted distribution for $. Recalling that a statistical test was used to show that the EDG data probably does not come from a homogeneous population, it can be seen that an analysis that employs the lumped data approach does not recognize the significant performance differences among plants and also overstates the knowledge , concerning the EDG failure probability at Plant 1. J Table 5 - Sununary of Example Results for 4, 1.0 05 kl.0 50 hl .0 95 1 a - Classical 0* 2.1 x 10 2 m 0W b - Single-staae Bayes 1.4 x 10-5 1.6 x 10~' 1.4 x 10-2 3.5 x 10-8 c- PEB: MI II 4.7 x 10d 4.4 x 10-8 1.7 x 10 2 5.9 x 10~8 d - PEB: MMMM 2.4 x 10 d 3.5 x 10-8 1.5 x 10-2 5.1 x 10-8 e-Two-stane Bayes 1.2 x 10" 3.3 x 10-' 1.8 x 10 2 5.2 x 10~' tunned Data" 1.4 x 10-2 1.9 x 10-2 2.6 x 10-2 2.0 x 10-2 Ncues: a) Imer bound,90% conndenceinterval b) Upper bound,90% coundenceinterval c) Maxhnum iuhand emimw 45
0- ... a d x ' g }; > a h t{, " e i ,o 30 ) l l l
\
} -6 k
*h+
. re 40- -
-1 ,
'e ',. log a log $
2 8 Hgure 'l - Posterior Distribution for (a,$) 200 A - -- j 1-stage MI. ell 150 - k 4 .---- lumped data 4 e 2. stage (prior) 4
- A 2-stage (posterior) 100 -- k
/%
t= * / '\
- g !
i i,A / g 50 - '.A ' . f \ d'A.' I.6 ' . . ',
'-/ . ... ,.,.___ 1 ' ;t- .
i : a , 01 0.02 0.03 0.M 0.05 Figure 8 - Comparison of Posterior Distributions for EDG No.1
I t
- 7. IMPLEMENTATION NOTES AND OTHER SPECIAL TOPICS With the development of a prior distribution and a likelihood function, the estimation process is fairly straightforward for problems of common interest in PRA; it often requires only the solution (analytical or numerical) of Bayes' Theorem for a single unknown parameter. This ..
section briefly discusses a number of topics relevant to Eq. (5), including some common pitfalls. It also introduces an advanced topic relevant to current PRA applications: the treatment of uncertain I data. i l 7.1 Numerical Solution Methods Except in the case of conjugate likelihood-prior pairs, Eq. (5) cannot be solved analytically. Numerical methods are therefore generally required to: a) evaluate the integral in the denominator of Eq. (5), and b) evaluate posterior distribution moments and percentiles. If there is a single unknown parameter (i.e., if G is a scalar), the required numerical integrations are easily performed using elementary quadrature schemes, e.g., the trapezoidal rule. These schemes can be implemented in a straightforward manner using spreadsheet or equation solving software. The latter often include automatic integration schemes which, in principle, allow the analyst to numerically evaluate the integral in question without worrying about the numerical algorithm. However, some thought is still required to properly employ these software tools. Some helpful suggestions are listed below. In the case of distributions with singularities (e.g., a beta distribution with a < 1 has a singularity at 4 := 0), many simple integration rules will not work if the analyst cannot exclude the singularities. In such situations, quadrature schemes which don't require evaluation of the integrand at the end points of the interval (e.g., the Newton-Cotes open integration, Gauss-Legendre, Gauss-Laguerre, or Gauss-Hermitian formulas) may be required. These schemes are described in numerous texts (e.g., [45,46]). Even in the case of automatic integration software, the analyst may be required to pmvide values for the effective upper and lower bounds of the integral in order to assure proper convergence. In determining these bounds, it is useful to first plot the integrand to better understand where most of the integral's mass will fall. Careless selection of the bounds can lead not only to computational inefficiency but also incorrect results. Deterministic quadrature schemes can be used when 0 is a vector and its dimension is small (< 10). When dealing with higher dimensional problems, or when dealing formally with numerous fuzzy data points (see Section 7.3), Monte Cado integration methods may be needed. The general Monte Carlo approach is described in numerous references, such as Davis and Rabinowitz [46). Applications to advanced Bayesian estimation problems which require the use of such techniques as transformation of variables and importance sampling are discussed in the literature (e.g., [47,48]). The use of these methods to treat fuzzy data points is discussed by Siu [49]. In recent years, Markov Monte Carlo methods (including the Gibb's sampler) have been i increasingly used to treat high dimensional problems. Gelman, et al. [25] and Gilks, Richardson, and Spiegelhalter [50] provide an introduction to these methods. 47
7.2 Cautions and Potential Pitfalls Sections 5.1.4 and 5.2.4 provide a number of cautions on the development and use of informative and non-informative prior distributions, respectively. This section discusses two additional areas of caution arising during practical applications of Bayes Theorem. .f. 7.2.1 Robustness of Solutions
'Ibe Bayesian estimation process is, in principle, a once-through process; the analyst develops a prior distribution, constructs a likelihood function, and computes the resulting posterior distribution. In this context, the notion of " robustness" of results is not particulady relevant; if the prior distribution accurately reflects the analyst's beliefs, and if the likelihood function accurately reflects the data generating process, the question of results variability with varying inputs /modeling assumptions does not arise.
In practice, of course, modeling approximations are made, and the issue of robustness i=== more important. Modeling approximations arise in the constmetion of the prior distribution, since the analyst generally wishes to spend as little effort as possible in defining prior distributions that precisely represent his/her beliefs. Noninformative prior distributions or simple parametric informative prior distributions are often used towards this end; the approximations undedying their use are re-examined only if they lead to unlikely results or if they involve dominant risk contributors. Modeling approximations that occur in constructing likelihood functions are somewhat less noticeable. Some approximations are associated with basic modeling decisions, for example, the assumption of a constant failure rate (leading to a Poisson or exponentiallikelihood function). Others are associated with the treatment of nonempirical data (e.g., expert opinion), for which the underlying generating process may not be well understood. Because approximations are made,it is clearly important to review computed results for reasonableness, and, in the case of particular significant approximations, to investigate the sensitivity of the results to these approximations. Two particular instances are worth noting:
- In cases where data are sparse, e.g., O failures in n demands (where n is relatively small),
moninformative distributions must be used with care. The particular form chosen [e.g., 3 flat, proportional to 1/$, proportional to $*3(1-4)*3] and the numerical cutoff bounds employed [the effective upper and lower bounds in Eq. (5)] can significantly affect the results. .it may be worthwhile to develop an informative prior distribution.
- In cases where data are less ep=e. noninformative prior distributions can lead to posterior distributions that are less sensitive (if so:newhat broader) to the exact form of the prior than those resulting from informative prior distributions. As shown in Figure 9, an informative prior distribution can " filter out" the likelihood function. This increases the sensitivity of the posterior distribution to the location and spread of the prior. Noninformative prior distributions, on the other hand, do not do much filtering, and therefore generate fairly robust results.
l 48
, . '. . *
- p e 7
non-informative g prior mmvvi ... function w x-0 Figure 9 -Influence of Prior Distributions on Robustness 7.2.2 Dimibution Figing in the numerical example of Section 3.2, it was assumed that the prior distribution was gamma. This resulted in an analytical solution to Bayes' Theorem; the resulting posterior distribution was also gamma. Now suppose that the computer code being used to propagate uncertainties through the PRA models will not handle a gamma distribution, but it will handle a lognormal distribution. In such cases, after carrying out the Bayesian updating as above, the analyst may decide to fit a lognormal distribution to the gamma posterior. In other words, the following approximation might be made: 1 1 finA ,2-exp< ! , -=< <m,o>0 xi(AIE) 42xoA 2,6 o / (63) This approximation is commonly implemented by equating the moments of the lognormal distribution to the moments of the posterior distribution, i.e., solving the following equations for and a: E[ A] I - eM +b' If 2 (64) 2 2 Var [A]- -ep+o{eo _1) Solving these equations using the data of Section 3.2 gives = -11.6 and a = 1.4. The 95th percentile resulting from this approximation is 9.5 x 10-5/hr, which is not too different from the exact value. The 5th percentile resulting from the approximation is 9.2 x 10/hr, which is almost 6 orders of magnitude larger than the exact value. This difference is due to the poor fit between the lognormal and the gamma in this example. (In cases where a' > 1, the fit can be quite reasonable, as the shape of the gamma distribution is no longer reverse J-shaped.) In situations where only the upper tail is of interest, as is commonly the case in PRA applications, the " quick 49
. :. l . : , L and dirty" fit of the lognormal distribution is usually good enough. However, if the lower tail b of interest, the lognormal approximation must be used with caution.
This problem with the method-of-moments fit can arise somewhat in reverse of the above, as well. Consider the following example. Generic data suggest, that for a certain initiating event .l-ofinterest, a mean value of 0.02/yr is reasonable. The data further suggest that the epistemic uncertainty about the true value of this parameter can be modeled with a lognormal distribution having an error factor (ratio of 95th percentile to median) of 14. This lognormal distribution is adopted as the prior distribution in the PRA. Data for a specific plant find no occurrences of this initiator in 28 years of operating experience. This data, O occurrences in 28 years, is to be combined with the prior distribution through Bayes' Theorem to obtain an updated estimate of the initiating event frequency. Because the lognormal prior is not a conjugate prior for any likelihood function, the integral in the denominator of Bayes' Theorem must be evaluated numerically. To avoid having to do this computation, suppose the analyst decides to use the method of moments to fit a gamma distribution to the lognormal prior. Using Table 1, one finds ot = 0.083 and p = 4.17 yr. Using Eq. (7), the posterior mean is found to be 2.6 x 108/yr. Is this answer reasonable? 12t us check our intuition. According to our prior beliefs about the frequency of this initiator, the mean recurrence time , is 50 years. We have watched our facility for 28 years and have seen no initiators of this type during that time. Based on our Bayesian update with this data, we now are claiming that the mean recunence time is 355 years! This appears to be an overly strong conclusion, given the available evidence. The problem is that the gamma distribution we are using to do the update is quite different from the lognormal prior we started with. The lognormal distribution is unimodal, and goes to zero as the initiator frequency goes to zero. The fitted gamma distribution, on the other hand, is reverse J-shaped, and goes off to infinity as the initiator frequency approaches zero. The lognormal prior expresses a belief, prior to collecting the plant-specific data, that the initiator d frequency lies in the interval (4.0 x 10 /yr, 7.7 x 10-2/yr), with 90% probability. However, the fitted gamma prior expresses a quite different belief: the 90% symmetric probability interval for the initiator frequency is (3.0 x 10'"/yr,1.2 x 10~ /yr)! This problem can be avoided by using other methods to develop a gamma distribution that approximates (" fits") the lognormal prior, such as preserving the 90% symmetric probability interval. However, with today's computers and software, h is a relatively simple matter to integrate the lognormal prior numerically. Doing the calculation numerically gives a posterior mean frequency of approximately 7.1 x 10~'/yr; this corresponds to a 140 year mean recurnace time. While this value is still gitater than the observation period, it does not represent as large an extrapolation of the current experience as the result obtained using the gamma approximation. 7.3 Treatment of Data Uncertainties The discussion in most of this paper assumes that the empuical data used in parameter estimation apply without question to the process being studied, that is, there are no uncertainties in 00 1
I the data. This assumption is actually not correct in a number of PRA applications. As one obvious example,it can be questioned whether generic industry failure data (or, for that matter, data from outside of the industry) are directly applicable to the plant being analyzed. In particular, should data from common cause failure events at one plant be used in the analysis of another, given that common cause failures are often the result of highly plant-specific system and process details? As .. - another example, the applicability of failure data preceding a major event (e.g., TMI-2) might be questioned if the event has led to significant changes in industry practices and procedures. This is not to say that data collected from a variety of sources should not be used; the point is that the use of such data requires a modeling decision, and that there are uncertainties inherent in such modeling. The subjective interpretation of probability provides a natural means to formally deal with
" fuzzy data," that is, data whose applicability is uncertain. This interpretation allows the quantification of uncertainty in individual data points using probability distributions. These distributions then make up the evidence to be used in the Bayesian updating process.
In our standby pump demand failure rate probiem, for example, assume that the number of failures is uncertain, and is characterized by a (discrete) probability distribution p(r). (Note that p(r) is assessed by the analyst based upon available information, e.g., event narratives, qualitative engineering information about the plant / system / component being analyzed). As suggested by a , number of papers (e.g., [6,52]), the posterior distribution for $ is obtained using I nt($Ip(r),n)- f n:($ Ir,u) p(r) (65) < r- o where n i($lt,n) is the posterior distribution that would be obtained from a conventional application of Bayes' Theorem, given that r failures have occurred. As noted by Martz, Kvam and Atwood ) [34), uncertainties in the number of demands can be treated in a similar fashion. [Ref. 34 also notes that Eq. (65) ignores sampling variations in p(r), but accepts the equation as a practical approach.] Because of the extra computation required by this approach, an ad hoc procedure commonly used in PRA applications for dealing with data uncertainties averages the data and employs the results in Bayes' Theorem (see, for example, [51]). In the above example, this results in an approximate posterior distribution, x ($lf ,n), where 3 f - } r p(r) (66) r-0 As discussed by Siu [49] and Siu and Mosleh [52), the " data averaging" approad ,vhile not strictly correct, yields results reasonably close to those obtained using the posterior averaging approach for a number of PRA problems. Possible problem areas involve: a) very large uncertainties in the data, or b) strong state-of-knowledge dependencies between the fuzzy data ' points, and may require investigation. I ; 51
A potentially more significant problem concems the basic framework for dealing with fuzzy data. The question is if data uncertainties should be dealt with in the likelihood function or via the posterior averaging approach described above. We note that we have actually presented a likelihood-based approach in Eq. (24) to treat situations where the datum is "known" to be bounded without comment. Both appror.ches, when considered separately, appear to be . reasonable,yet they do not appear to be mutudly censistent. Additional fundamental work on the f nature and treatment of uncertain data appears to be needed to resolve this question. I 52
l
?
I
- 8. CONCLUDING REMARKS To many PRA analysts, " Bayesian analysis" can mean a mechanical updating of either a roughly defined informative prior distribution (i.e., a prior distribution which only roughly approximates the analyst's belief concerning the parameter ofinterest) or some fonn of " objective" .. )
prior distribution (e.g., a non-informative or maximum entropy prior). While Bayes' Theorem is straightforward in principle, and while practical Bayesian analysis for PRA applications is often a i fairly simple process, there are situations for which a mechanical approach can lead to incorrect or ' misleading results. Examples of such situations include: cases where the data are sparse (this increases the importance of carefully defining an informative prior distribution), cases where the I prior distribution is too nanow (i.e., the prior overstates the analyn3 knowledge about the parameter), and cases where the data are uncertain. We hope that the combination of formal background and practical tips provided in this tutorial and in the references will help the PRA analyst identify and appropriately address these problems when perfonning a Bayesian analysis or when reviewing the results of such an analysis. We recognize that some of the formal methods discussed in this tutorial can appear to be quite (and even perhaps excessively) complicated to many analysts. While it is true that the methods generally require more computation than those required to develop classical point estimates for typical PRA model parameters of interest, we believe that the benefits from these methods (e.g., the more realistic treatment of the impact of industry evidence, the quantification of parameter uncertainties in a format for subsequent use in a PRA model, and the requirement on the analyst to think more carefully about the exact problem to be solved and the evidence used in the solution) generally outweigh this objection. Of course, it is recognized that great amounts of effort are not warranted if the careful characterization of uncertainty in e. parameter does not affect the decision to be supported by the PRA. We observe that most of the computations discussed in this paper can be performed using widely available spreadsheet or equation solving software packages; this objection should therefore be far less important than it was a number of years ago. We anticipate that with continuing improvements in computer hardware and softwue, limitations in computational resources will decrease in importance as a controlling factor on the analyst's ability to formally assess the uncertainties in the parameters of increasingly realistic (and complex) PRA models (e.g., for system repair / recovery times, for regression-based treatments of human error probabilities). 1 As a final observation, we note that since the objective of Bayesian estimation is to develop a posterior distribution for a set of uncertain parameters 9,, this development has been the focus of l our paper. It is assumed that the posterior distribution can be used to generate the information i needed by a decision maker, how the decision maker should use the information is the subject of the field of decision making under uncertainty [1]. However, it is worth mentioning that decision i makers often do not employ the entire posterior distribution, and that the problem of summarizing the information content of the posterior distribution can go beyond a simple statement of moments or percentiles. O'Hagan [53] notes that the development of a prior distribution employs summary information (e.g., indications of central tendency and distribution shape) and that this same !
" language of beliefs" should be employed in the summarization of the posterior; probability distributions are used as intermediaries because they allow the use of mathematical analysis tools.
In such a framework, methods used to appmximate general posterior distributions (e.g., [25,54]) 53 m .
s . ', ..;., can be viewed not only as aids to numerical analysis but also aids in understanding and using the results of the analysis. We also note that the decision maker may not even be directly interested in 8. It may be of greater interest to determine the probability that an observable random variable, for which 0_ is just ' a characteristic, takes on a particular set of values. In the standby pump problem, for example, the decision maker might like to know the pmbability that the number of failures (r) is greater than some fixed value (ro ). This probability is determined by averaging the aleatory distribution for r over the epistemic posterior distribution for the demand failure probability 4, and then summing this averaged aleatory distribution over the interval ofinterest. The pmbability that r is greater than roin a given number of trials (n)is given by
.i P{r>ro I n} - ] fr!(n - r) $'(1-4)"~'ai(4 I E)d$ (67) r.,,,o Eq. (67) is an example of a " predictive distribution;" it provides the probability of a future / hypothesized event. This pmbability, of course, is the type of input needed by the decision making process illustrated in Figure 1. Bayesian estimation simply serves as a tool to formally incorporate, through the posterior distribution for $, relevant evidence (E) into the decision maker's probability.
l l 54 l 1
,:,.',9.e '
s ACKNOWLEDGMENTS The authors gratefully acknowledge the helpful comments of G. Apostolakis, A. Mosleh, T. Leahy, and the anonymous referees, and the assistance of D. Siu in preparing the manuscript. This paper was supported by the Idaho National Engineering Laboratory Center for Reliability and ,,.. Risk Assessment under the U.S. Department of Energy (DOE) Idaho Field Office Contract DE-AC07-76ID01570. The opinions, findings, conclusions, and recommendations expressed herein are those of the authors and do not necessarily represent the views or official policies, either expressed or implied, of the U.S. Nuclear Regulatory Commission or the DOE REFERENCES
- 1) R. L Keeney and H. Raiffa, Decisions with Multiple Objectiws, Cambridge University Press,1993.
- 2) G. Apostolakis, " Probability and risk assessment: The subjectivistic viewpoint and some suggestions, NuclearSafety, 19, 305-15(1978).
- 3) G.W. Parry and P.W. Winter, " Characterization and evaluation of uncertainty in -
probabilistic risk analysis," Nuclear Safery, 22, 28-42(1981).
- 4) G. Apostolakis, 'The concept of probability in safety assessments of technological system s," Science, 250, 1359-64(1990).
- 5) S. Kaplan and B.J. Garrick, "On the quantitative definition of risk," Risk Analysis,1,11-37(1981).
- 6) A. Mosleh, " Hidden sources of uncertainty: Judgment in the collection and analysis of dsta," Nuclear Engineering and Design, 93, 187-98(l 986).
- 7) G. Apostolakis, "A commentary on model uncertainty," in Model Uncertaintyt Its Characternation and Quannfcation, A. Mosleh, N. Siu, C. Smidts, and C. Lui, eds.,
Center for Reliability Engineering, University of Maryland, College Park, MD,1995, pp. 13-22.
- 8) B. De Finetti, Theory of Probability, Wiley, New York,1974.
- 9) D.V. Lindiey, introduction to Probability andStatisticsfrom a Bayesian Viewpoint, Part 2:
h iference, Cambridge University Press,1%5.
- 10) R.L Winkler and W.L Hays, Su=dc+n Probability, inference, and Decision, 2nd FJuion, Holt, Rinehart and Wm' ston,1975.
- 11) D. Kahneman, P. Slovic, and A. Tversky, Judgment Under Uncertaintyt Heuristics and Biases, Cambridge University Prere,1982.
55
~ s .a,', , . c '.
- 12) H. Jeffreys, Theory of Probaliitity, 7hird Edition, Clarendon Press, Oxford,1%1.
- 13) G. Apostolakis, S. Kaplan, B.J. Garrick, and R.J. Duphily, " Data specialization for plant specific risk studies," Nuclear Engineering and Design, 56, 321-9(1980).
- 14) W.E. Vesely, "More comprehensive appmaches for determining risk sensitivities to component aging," Nuclear Engineering and Design, 106,389-97(1988).
- 15) A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York,1%5.
- 16) G. Apostolakis and A. Mosleh, " Expert Opinion and Statistical Evidence: An Application to Peactor Core Melt Frequency," Nuclear Science and Engineering, 70,135-49(1979).
I
- 17) M. West, "Modelling expert opinion," in Bayesian Statistics 3, J.M. Bernardo, M.H.
DeGroot, D.V. Lindley, and A.F.M. Smith (eds.), Oxford University Press,1988, pp. 493-508.
- 18) A. Mosleh and G. Apostolakis, "The assessment of probability distributions from expert opinions with an application to seismic fragility curves," Risk Analysis, 6, 447-61(1986).
- 19) H.F. Martz and M.C. Bryson, "A statistical model for combining biased expert opinions,"
IEEE Transactions on Reliability, R-33, 227-32(1984).
- 20) S. Kaplan, "On a 'two-stage' Bayesian procedure for determining failure rates," IEEE Transactions on Power Apparatus and Systems, P AS-102,195-262(l983).
- 21) JJ. Doely and D.V. Iindley, " Bayes empirical Bayes," Journal of the American Sa**al Association,76, 833-41(1986).
- 22) K. P6rn, 'The two-stage Bayesian method used for the T-Book application," Reliability Engineering and System Safety, 51,169-79(l996).
- 23) V.N. Dang, T-L. Chu, and N. Siu, "lognormal extension of the two-stage Bayesian snethod: Applications from the Surry shutdown study," Proceedings of Probabilistic Safety Assessment international Topical Meeting (PSA '93), pp.1353-9, American Nuclear Society, La Grange Park, IL,1993.
- 24) M. Kazarians and G. Apostolakis, "Modeling rare events: the frequencies of fires in nuclear power plants, in Low Probability-High Consequence Risk Analysis - Issues, Methods and Case Studies, R.A. Waller and V.T. Covello, eds., Plenum, New York, 1984, pp. 33-53.
- 25) A. Gelman,J.B. Carlin, H.S. Stern, and D.B. Rubin, Bay 2dx Iksa Analysis, Chapman and Hall, London,1995.
56
, .' 3. , 9. o t
- 26) J. O. Berger, S*nd&d Decision Theory and Bayesian Analysis, Second Edition, Springer-Verlag,1985.
- 27) B.P. Carlin and T.A. I.nuis, Bayes and Envirical Bayes Methods for 1kaa Analysis, Chapman and Hall,landon,1996.
- 28) J.S. Maritz and T. Lwin, EnpiricalBayes Methods, Second Edition, ch= aman and Hall, landon,1989.
- 29) W. Galycan, J.P. Poloski, and G.M. Grant, " Emergency diesel generator power system seliability, 1987-1993," Proceedings of Probabilistic Safety Assessment international TopicalMeeting (PSA '96), pp.1809-16, American Nuclear Society, la Grange Park, IL, 1996.
1
- 30) H.F. Martz and M.C. Bryson, "On combining data for estimating the frequency of low l probability events with application to sodium valve failure rates," Nuclear Science and Engineering,83,267-80(1983).
- 31) J.K. Shultis, D.E. Johnson, G.A. Milliken, and N.D. Eckhoff, "Use of Non-Conjugate Prior Distributions in Compound Failure Models," NUREGlCR-2374, U.S. Nuclear Regulatory Commission, Washington, D.C.,1981.
- 32) H. F. Martz and R. A. Waller, Bayesian Reliability Analysis, Reprint edition with .
corrections, Krieger Publishing Co., Malabar, FL,1991. ! l
- 33) R. E. Kass and D. S!rffey, " Approximate Bayesian inference in conditionally independent Mm2M wiels (in,wi,etric Empirical Bayes)," Journal of the American Statistical Asscciation,84, (1989).
- 34) H.F. Martz, P.H. Kvam, and C.L Atwood, " Uncertainty in binomial failures and demands with applications to reliability,"InternationalJournal of Reliability, Quality and Safery Engineering,3,43 57(1996).
- 35) P. K. Samanta, I. Kim, S. Uryasev, J. Penoyar, and W. Vesely, " Emergency Diesel I Generator: Maintenance and Failure Unavailability, and Their Risk Impacts," NUREG/CR-5994, U.S. Nuclear Regulatory Commission, Washington, D.C.,1994. i
- 36) P. Brdmaud, An A.troduction to ProbabilisticModeling, Springer-Verlag,1988.
- 37) E.T. Jaynes, " Prior probabilities," 1EEE Truruactions on Systems Science and Cy6ernetics, SSC-4, (1968).
I
- 38) C.L Atwood, " Constrained non-informative priors in risk assessment," Reliability Engineering and System Safety, 53, 37-46 (1996).
1 57
~
o /, .' .N 1 i ;
- 39) T.A. Wheeler, et al, " Analysis of the LaSalle Unit 2 Nuclear Power Plant: Risk MethM ;
Integration and Evaluation Pmgram (RMIEP), Parameter Estimation Analysis and Screening Human Reliability Analysis," NUREG/CR-4832, Vol. 5, U.S. Nuclear Regulatory Commission, Washington, D.C.,1993.
- 40) R.C. Bertucio and J.A. Julius, " Analysis of CDF: Surry, Unit 1 Internal Events,"
NUREG/CR-4550, Vol. 3, Rev.1, , U.S. Nuclear Regulatory Commission, Washington, D.C.,1990.
- 41) P.K. Goel, " Software for Bayesian analysis: cunent status and additional needs," in Bayesian Statistics 3 J.M. Bernardo, M.H. DeGroot, D.V. Iindley, and A.F.M. Smith (eds.), Oxford University Press,1988, pp.173-188.
- 42) C.A. Claro'tti and WJ. Runggaldier, Response: On the 'true' value of a failure rate and related statistical issues," Reliability Engineering and System Safety, 26, 25-34 (1989).
- 43) G.E.P. Box and G.C. Tiao, Bayesian' Inference in Statistical Analysis, Addison-Wesley, Reading, Massachusetts,1973.
- 44) U. S. Nuclear Regulatory Commission, "PRA Procedures Guide," NUREG/CR-2300, Washington, D.C.,1983.
- 45) B. Carnahan, H.A. Luther, and J.O. Wilkes, Applied Numerical Methods, Wiley, New York,1%9.
- 46) PJ. Davis and P. Rabinowitz, Methods of NumericalIntegration, 2nd FAition, Academic Press,1984.
- 47) L Stewart, "Multipio-rar Bayesira in/erence - some techniques for bivariate analysis,"
in Bayesian Statistics 2. J.M. Berm.rdo, M.H. DeGroot, D.V. Lindley, and A.F.M. Smith (eds.), Elsevier Science Publishers,1985, pp. 495-510.
- 48) H.K. van Dijk and T. Klock, " Experiments with some alternatives for simple importance sampling in Monte Carlo integration," in Bayesian Statistics 2, J.M. Bernardo, M.H.
DeGroot, D.V. I.indley, and A.F.M. Smith (eds.), Elsevier Science Publishers,1985, pp. 511-530.
- 49) N. Siu, "A Monte Carlo method for multiple p=-r.cter estimation in the presence of nacertain data,' Reliability Engineering and System Safety, 28, 59-98(l990).
- 50) W.R. Gilks, S. Richardson, and DJ. Spiegelhalter, Markov Chain Monte Carlo in Fincrice, chapman and Hall, London,1996.
- 51) K.N. Fleming, A. Mosleh, and R.K. Deremer, "A systematic procedure for the incorporation of common cause events into risk and reliability models," Nuclear Engineering andDesign,93, (l986).
58
.' , , . ', . p. o *
! 52) N. Siu and A. Mosleh, " Treating data uncertainties in common-cause failure analysis," l 1 Nuclear Technology, 84, 265-81(1989).
- 53) A. O'Hagan, " Shoulders in hierarchical models," in Bayesian Statistics 2, J.M. Bernardo, ,
M.H. DeGroot, D.V. Lindley, and A.F.M. Smith (eds.), Esevier Science Publishers, 1985, pp. 327-344.
- 54) C.N. Monis, ' Approximating posterior distributions and posterior moments," Bayesian Statistics 3 J.M. Bernardo, M.H. DeGroot, D.V. Lindley, and A.F.M. Smith (eds.),
Oxford University Press,1988, pp.173-188. l i 1
)
1 { l l l l l l l i l l 59 l
A T A D E H L T A R N F O O O I F T E A S E R N S T E O E G T P I N N I O S E I M S V T M F I I E O P O M T E C U M C M O N O E E R 7 T O C F T V G 9 T I F N I 9 E T Y E E T S 1 R A R M U R R U O L E C E , A L T A G E N 0 B A A N A X W 3 V L O N E O J E U 1
. I A R G T M E W E D D E A H B R N R Z R T & O A A I O T H R N I O B C C S A A N T O I I E G E E R S L R S N H Y C O O T L U C I A N F R T N O N A A S T
\ T N E N E H E S T U M E -
S R R _ S P O _ E F S . S E A C _ I F _ F O
R S D ", O R E I' O C N N T A O E A L I S C P T I A E D E T H N B C T I E D P F R L X O U E O F O H E E S C L W D N B E E A ". _ A I L T V N M R . T L A R E U E A L ", O N R R C F C P S R N _ S E T A - E A N E M V V E P .
- I A R ", M F T -
S H E S O N G N H G S E N E A T N E L T I H D E S D I E S E N D S V A R A H ", W A A E I S E P R T E L N L M O GS L N O _ O M NI E E C C E IL D V L M G N I B A T DH A R A 2 N A D E S UC D N N T D T JT b - A I S N A u T E E RW r N T . I T M O G E S . S AS FE R N E N V E O I S R H I E C TE I ST N D N S I C U I A N E NO SM U
- Q E O I _
IR E AO M N T E P R BR E A I V S F , C L - I AG F A E N ) I T HNI F EE D V A S C C A I I E A E N ET T DV T L ( F J O E S IO C E H B I VM E R S Y O S M F OE J I F S O RR B E L I I T P O C B T M N R " U A N E S NO D T E MM E A E E S D OE I CN R R E I CG R: " O A " " " A ET EN SA HA H TM AT .
l Il l {ji l! l l ,i DD SO NI ET AD S
" UR M SN AE E EE CD H GT R T NF LO AO A E H NN T CS OI A T I P LN TS I EE AR C NM RO I NS ET T OS PC N SE OA A RS F ES D N PA NL A AAN H DE T E S" TO T ES NI R H HN ET Y
D E H G I TA L MA U T L ,P ER T A HR GE S R G ET AP I NO N S H VN EE A MD E S M E S WM OE N R L E HV GA E B I O N D O R R I T 3 N R A .P S N A P M SM S EM M E E R O U S A NI S G S U T LF A H T T E PO AN V TS A R C I M M A A O E rNT u EL RG R cMU E ES FN I" O VE X OR Y SE T E R G SC E EN \ S R MPE T SA A SM A M MH R E S IT T AR O P ESE S OF P H' T TR A TTA N E E NU O HSP C FEL I TT R OMA T NE N EV A TEV
- YGE D FVO S E NA N IER G H ANT E H P N T MAO M SFM I " " MN M " OI e D O N C
_ I E F R ll
D b3. $m E k _ T i _ e cs A nr ao t s . R EL mt rc a l u s e G D oi fd rn R E eI P TO : ss s NM I i t ne one av E N C ri ec pe t s E e Off E e SN r u s m 4 RA i a t s ns T e EM M ee mn e ve DR gi atc NO n affe ME AF : R R cs ir E mto U H P da nc oi cdn T EI R k A e f b a $ g em A
T C TEN E N R NHA C
' E N" ET N T M M T A E ,A EGA M P ER GN R
_ M CG AI S O O NO C AR MP NSM" AS E S F R MI L E R MB S P F ON O FO T OE _ RI NSR AI P D R D N ET L D A E PC A C E PC E Y N YP RCO C"S A T TS EN SN N A T N S E E R E ,M TC FI ESR R O A E P ST SSO EP P O M R M I OA N NE FA EC R E O P TL CO E H CN E T P IR LP PC SO H N S ' I - T E EN YD TT _ M T HO E I TI FTNEA NC EE F MP Y u O G A NI FCT AR SE E S u N OE I GN r AS P S AI E N S MR YS' CN SA R"E AS 5 F O TE A NI N M'C O T O W C E A AN C AE FR G A M EA TT EN LR PA ON CF PA TF ER N ML RO AT OP ES NO E HM AF TN R C MT E RE G TEL SN OM EN S P PE HIN B SE . EN R MG TO O O EY I A G R SO GT N G T P SP AE RA NNC N AM NF AM IOA E E O AA E T H C MS H LS O CSE AEOW UHT TY CT ES LT TR R HS A YE AO HT NT TA VFTG N TAD N EOIA O E ET NT I EN RM TO : TTUS T RA OE IN N ORT A IM IG P O NARY D U
; NA SS I POL N QA S EN EE S Y PR E E T SA DO U BSPA M RE L " M " D L "AOE M " B .
U C O S N C E O E R C R
J SSDES SNSS IAEHS SORI HSTE EIO T A N NTMT N TBNE OV EA VU N E I E I S E M C IL T M E
,T TA H E GE .NDC CV G G AC MAFANE EE F
I AN NN R F S AA ONE FEN A EC I M MM F G N R R IT S O E SU TA H N M SC E R F P EO E R UI M O OH E PE T DBA E F IW S CD GRR N Y U NS A E AN E T DANO C N A P VI EN S I LA F O NI C OS M TE E FNT C S OTL IU OAA N O F O ALTAC TPLLU TA A RN N A S G NEEO O E OCPC C IIO MNE IN ETV PE T NTT W ORLY E AACYA LRLF S MRATR EUII N R ERID OPPL VCWT E N 6 O FO P S R SCT IO DEH YGD E ES' A E E P' FF
" H T O I _
R EF ,T T L - EO O E EAFTS N C E GVD T _ R H STOL T NS M NO O Y N E UE SS G IR TPTH L C EAE A AME LA TI AL HER TR A N GIM I CI H ANM TO TE , O ST ST H E IA S T NLT S SLI VS HM EN IAA CEE 'E NL TE EI IO T S ". R MNP S O YE GLEISN AAR NE S TY.SVI I T NIC FTGDT C NDO I FNONC R ENIFL AELAE N UMATO C E TMO F R ,ESH S SOLE SDYF ETNPIT EEHE HSESS HSTMT TNENAF TSEIO
" IBIBO " AMTN .
4 N O I t n e e c e c e c T m nt aa n s a d i md s A s s e mlp r om mn r e or o ore n n M s s fr e eT f rT e E cT R A P P O .
- F N
I T N E _ M e e c e cs cs S ia t v n a88 nr ao ir mo _ S mt oa t ct ea mU a jD r8 r c nc E b O fo8 r e 1 foid r eI n oi cdn S P P Ei S ll A .' . ' E C _ N s A M n s t R i os t e s n o s s n o n o n e s t y g yn s mn i O i t t t l t ac i nr e i a eo l ia hirt vpo t F mr r u c e t a g g ci bt aDa t ao nrp MoepRe i t r o c t Ee i p oo R e s fA l e R fS n I s n l l A e v n E n R O - E i i P
- L
) 6 .
R _ e g a s 9 4
. En T r e
v 6 9 Q- _ A t y r 3 6 E, s 6 ( d u - 9 2 s e r En in u . 6 . e L _ - 9 1 l ia F g a
. E-E 5 s m
r e v
. In D - 9 4 t e
s A y r O _ > 5 9 3 S y y t d s/ u I-M 5 9 t f S e a J n En D 2 N _ m 5 9 1 A.E- . E _ vs 4 9
- En R 4 T _ vr 4 9
3 8 6 4 O E-E 4 8 C A_
- 9 2
N _ = 4 9
- $n A 1 1
3 M _ 9 4
)
'$n R _ 1 3 9
t n c e . O 3 ' - r e ' E-F ld h o _ = 3 9 2 ( e p R s e r 3 t R e - In E s h T 9 1 e g a
- I-P t
2 u _ . 9 4 O _ 3 - 2 d e c m.- - En 0 _ 1 9 r o . - 1 3 F . -E- - T _ 1 2, 9 2 - En N t 2 A _ 9 . . 1 E ., L 8 6 4 2 0 0 0 0 o0 8 6 4 2 P s
T S N E E I M R E O V G O E R T P A M C S I E E U T S , A S E L E I N P T I M A s T L E L M S N T P M E E PE M S I M I R X . E R E CU I T T O G ST S _ S G A I L _ E A E N DU F _
/T C O C P T A _
N A M E A M C HD N M O O G I N O S A R R O T O F S T S F E N D RE A . T EC R S E I . E G U T VN T P N B M A OA N I I E L I E T D R H E L M 9 N N r T R EP A L I F r A D S RM EO E G P E G WC A F E Z N N A O C I I Y A N N D R M F E A A N S O S G E T E O M I I D A R R F R A U T B O O O L L N A F F G U C E T R D O E G N M A E N E I P D P A T T AR O N C O L N 0 M E , T E E 0 E C S V S 6 H E D E R 3 T R LV E D E E AI D D D D P PT I N E E IA C A I T 2 CI E F A 2 NT D R I G II U T E r RN F H N R u PI F T E G O A R D G B T A I A A S
G S E I R O G N E O T I A T C U 0 1 L E O T S A E L R S P S M D E E N N T A E , E V C N I N O T A , I C M T E R N N A F O O E G C F
\
F I C I I E R T N S F E I A E I L P D M D T A N R / N N L O O G E O A C F N D I N R I I T O L E R A I A P E M Z T I E E I A R N N L N R E A I B A E T M G O G P A U N R R te O M H E P O .
T N < 8d 3 E > M e c n e c n a s a S mn d r e r o S or fT r f r e E e P S j S e c n g n AS a mlaio ee r n i r ED n i rit o ed t n ig e ao n CO MC E e NH / m 1 AT 1 i l n T nio a n' M ME o" ia" t z otu lll ao l s e in 'I R f i e" tF d r n O O a F R, c E i s md o n P n or cT e E C R < N
S A E. T U A S D S . N I S O
" N A I O T T E I A A C T D G N A S E A G " E E L E _
M E M I L L S R L I R O A U O L P O R F A " G , R R E Y T O E , G T R N F - P S N A A E T W I C D M D . E S S N E E N I U E G E O C R A V T N C C R E E L E S A I N E O D S P R T L D A S F I A R N P N I N S B O R A M I L A E N F E L E F P O N L P T M Y F C O E " O A O I R T E S C L E T U S H H S S P E B A T T G T E Y R S M C R I N R O Y U O R U O E H E O T L S O R P T V T C B E L 2 F T E R I I A E A L A 1 N S R O W T L P B B I F C U S O I L E N L E G N R S L E D O Y L F E I P S I T I I D E F R O W A V T U W E O L P L O C T O T L S P R E S T L T I E R M P P I A E W T O E S T F N D C A T T O N O O E N S U A T I L " I T A R L C I S T A D O A I D M P E A L I T V D E O U Z E U A E N D R D S I R G C I N F E S N I L E M I A E F D L L T E R " G R O N I A N B O R E I W I I F T O W T C O R A " L L F N E T E H T U A F A T P T F S S C A N A S O O E I T I L E F D M- R R S F P U F N K - E - - M S A U C M E S T O A U T I S F L N
I) l ll1 il! l I i!tl iil ll Y R O G E T S A S T C R L O U
" T S S C E S A R E F N N E L O V A I I N T N T O S C O C I E E I E T I P T F A R S C F Z O N E E I G E I L N E T L L A E T D A M O A G T A E L O C N R A C N P T R 3 O O 7 M I M U F A 1 I 9 I E F E P T T D 9 L N E T N E A A N 1 C I D I O D Z A / F W I , E E S E G G R N T T R D E N N N O A N S U I I I F G E U T O R O K R R M G b T O T C R S O E U u G A E A " G A c P E A L F E E A O T T N C R F N L H A A S I N A O A : A S C D A A M E N K E F D W \
N C O R Y S R O I E O N N I O R M A U N I O E T W A I Y G T R A N P Y C E I M E Z P I F A P T N A F I U M E I C O A I R N N O I Z T I L U F G O A R L I N F E L E O C G G E N E F V A D R R R A D E E V P O O L P G I D E H - L - R - T - - S A A O S E D M E E R I S R T 1 !
l S S E M S E S L E B C _ O O . R R D S P P E D T S E N E T U N E C N S I M N E T E F S E M N B E S M S E D E R S M S O E S T T S F S S N E A R S E E Y E A S S _ M E P E S S T V S E S A E O I F C S S N T O N E H S C A . C T A S E R M . O I I J O R . R W S B T O M P S S U A F M D E S S C R S L E N E I E L T E N E D P , A A V E T N P I I V A I F L R C T I N / O A O O C T I E S F S E C M R W S u F E I U E , E A F F L S I R S E F E A V P A S E E E P B N O S M R O L L T E T N I A A V S G D A O T N N N I N N E I C O O E T O I T K T A I E A T I I D A O A A T B I T A R O M C Z A T A E G L R R I Z S I Z L E O N N I A N I T H F A N H I N A N S N E G A A I E I H R G S T G S R T O R I N R A G F G O S E O N N G L A M E D I A I N L F H E R E T T I I O P V W C G A N W M O T E U N U I E E R N I D I L M W L P A V N K A A O O T M L O A V X H R N I P S C T E E E - - I - - - E C C H E E R T R O N
- 8 9 9 1 G N I R P S
- 8 S 9 T 9 C 1 E
J , O R R E P M M H U S T S 8 E O 9 4 N 7 B 9 1 O 9 : 1 . T 9 R N 8 S 1 O O R E F I A E L , S E V I 3 P I Y I M O C T R H E R C 5
/ E S D A E 1
B K D R E M R N N I L E O O E D U V T W I L D O C S A T E N E D S C N H J N I E _ C F O A M F M S O R M O E W P D O G K E O C D A E I T I N N E V N R R E A W E E E O M R M P F
- E : O G D V T N N T N E O N O O I T R E I I N T A P M T T O E R M M A A I E G I O D T S M E C N N I T M E E V C N M C M M E I I S I M E R L - - L O L -
B B C P U U E M P P R I
Review Paper RADIOACTIVE MATERIALS IN RECYCLED METALS-AN UPDATE Joel O. Lubenau* and James G. Yusko t shutdownsnthat typically totalled about U.S. $10,000,(XX) Mstract-In April 1995, Health Physics published a resiew per event and, in one case, totalled U.S. $22,908.000 paper titled "Radioacthe Materials in Rec 3cled Metals." At (Lubenau and Yusko 1995). Although the majority of the that time 35 accidental meltings of radioacthe sources in metal mith wcre reported,includmg 221n the U.S., along with radioactive material found in recycled metal scrap is in 293 other esents in the U.S. where radioactise material was the form of contamination of the metal scrap by naturally found in metals for recychng. Since that date, there hate been occurring radioactive material (NORM), the metal reeY-additional accidental meltings of radioacthe sources in metal tim.g mdustry, and principally the steel manufacturing mills both in the U.S. and elsewhere. There also was an industry, is most concerned about sealed radioactive incident in Texas that imolved stolen radioacthe degices, sources or discrete radioactive sources including alloys which resulted in exposures of members of the general public. containing radioactive material. In addition to the costs Also, the U.S. Nuclear Regulatory Commission took steps to resulting from accidentally melting sources, the paper address the underlying problem of inadequate control and took note of the potential of such sources to cause accountability of radioacthe materials licensed by the Nuclear significant exposures to radiation for unsuspecting work-Regulatory Commission. The Steel Manufacturers Association ers and members of the public. NRC staff was directed made asallable data collected by its members begmning in 1994 that expanded the database for radioacthe materials by the Commission to explore the problem and, working found by the metal recycling industry in recycled metal scrap together with the Agreement States, to develop recom-to user 2,300 reports as of 30 June 1997. mendations for future actions, liculth Phys. 74(3):293-299; 19918 Key words: contamination: exposure, radiation; radioacthe materials; waste disposal DATA UPDATE Since publication of the 1995 paper,14 additional accidental meltings of radioactive material have been INTRODUCTION reported for a total of 49 (Table 1). Of these, ten occurred outside of the U.S. in Poland, South Africa, Bulgaria. Tm Amu.1995 review paper titled " Radioactive Mate- Canada, the Czech Republic, Austria, Brazil. Italy, Swe-rials in Recycled Metals" reported a database of 315 den and Greece. Many of these cases first came to the events in the U.S. for the period ending 31 December authors' attention as a result of conversations with 1993, where radioactive material was found in recycled knowledgeable persons at annual Health Physics Society metal (Lubenau and Yusko 1995). In 22 of these inci- meetings and at the 9th International Radiation Protec-dents, the radioactive material was not discovered until tion Association (IRPA) Congress in 1996 in Vienna, after the material had been melted with the scrap metal. Austria. Attempts are made to confinn oral reports in U.S. metal mills that melted radioactive material incurred writing, and only when this is successful are the data costs for decontamination,* waste disposal,' and mill added to the database. The authors have reason to believe that in some cases operating mills or government agen-
' U.S Nuclear ReFulatory Commission. Washington. DC 20555 cies are reluctant to provide information because of the (xxil; ' Pennsylvania Department of Environmental Protection. 400 negative perceptions that might result from discovery of Waterfront Drive, Pittsburgh, PA 15222-4745. the meltings.
1 Cleanup criteria that were applied usually followed those found m NRC Regulatory Guide 1.86 (1974L The Bulgarian incident resulted in *Co contami-
' See 621R 13176 for the NRC Staff Technical Position relative nated steel product (plates) exported to the U.S. where to the disposal or electric are furnace dusts contaminated with mCs the contamination was detected by a steel fabrication (19 March 1997). plant in Mississipp;. The South African incident was a For correspondence or reprints contact: J. O. Lubenau. U.S. melting of a '"Cs source, which was discovered in Italy Nuc r gulatory Commission. M/S 0-16Gl5, Washington. DC ~
in a load of vanadium destined for Austria. A melting of (Mamucript received 31 March 1997; rerned mamucript received 9 a '"Cs source at a steel mill in Quebec, Canada, in 1995, September I997, accepted 27 October 1997) 0017-9u78/98/53Mu Cop >nght O 1998 Ilealth Phyucs Society 8 Mill shutdown costs can he as much as US55003100 per day. 293 The copyright law provides that no copyright exists in works prepared by an officer or employee of the U.S. Government as part of his or her official duties.
294 Health Physica March 1998. Volume 74. Number 3 Table 1. Confirmed worldwide smeltings of radioactive sources. Entries in italics are additions to the 1995 table. No. Year Metal location Isotope GBg 1 -' Au unknown. NY 2
"Pb unknow n 83 Fe Auburn Steel. NY "'Co 930 3 83 Fe Mexico" *"Co 15 AX10 4 83 Au unknown. NY 2 Am unknown 5 83 Fe Taiwan * *"Co > 740 6 84 Fe U.S. Pipe & Foundry. AL ' "Cm 0.37-l.9 7 85 Fe Branl* *"Co unknown 8 85 Fe TAMCO. CA '"Cs 56 9 87 Fe Florida Steel. TN '"Cs 0.93 10 87 Al United Tech. IN 22*Ra 0.74 11 88 Pb ALCO Paciric. CA '"Cs 0.74-0.93 12 88 Cu Warriegion. MO Accel unknown 13 88 Fe Italy *"Co unknown 14 89 Fe Bay;u Steel, LA ' "Cs. 19 15 89 16 89 Fe Cuemp Spec. PA n unknown Fe Italy '"Cs 1.(XX) 17 89 Al Russia unknow n unknown 18 90 Fe NUCOR.UT "Cs unknown 19 90 Al Italy 20
'"Cs unknown 90 Fe Ireland 21
'"Cs 3.7 91 Fe India" *"Co 7.4-20 22 91 Al Alcan Recycling. TN Th unknown 23 92 Fe Newport Steel KY 24
'"Cs 12 92 Al Reynolds VA 22*Ra unknown 25 92 Fe Border Steel. TX 26
'"Cs 46-7.4 92 Fe Keystone Wire. IL 27
'"Cs unknow n 92 Fe Poland 28
' "Cs unknown 92 Cu Estonia / Russia *"Co unknown 29 93 Fe Auburn Steel. NY '"Cs 37 30 93 Fe Newport Steel, KY '"Cs 7.4 31 93 Fe Chaparral Steel. TX 8"Cs unknow n 32 93 Zn Southern Zinc, GA 33 DU unknow n 93 Fe Kazahkstan* *"Co 0.3 34 93 Fe 35 South Africif '"Cs < 600 By g ' '
93 Fe Flonda Steel. TN 36
'"Cs unknow n 94 Fe Auburn Steel. IL 37
'"Cs 0.074 94 Fe U.S. Pipe & Foundry. CA 38
'"Cs unknown 94 Fe Bulgaria * "'Co 3.7 39 95 Fe Canada" 40
' "Cs 0.2- 0. 7 95 Fe C:ech Republic "'Co unknown 41 96 Fe Sweden "'Co 87 42 96 Fe Aus>ria 43
"'Co unknown 96 Pb Brazil" 44
"Pb anknown 96 Al Bluegrass Recycling. KY '"Th unknown 45 97 Al White Salvage Co.. IN 46
A m unknown 97 Fe WCl. OH 47
"'Co 0.9 t ?)
97 Fe Kentuckr Electric Steel. KY ' "Cs 1.3 48 97 Fe Italx ' ' "Cs 37 49 97 1e Greece '"Cs unknown
- Multiple cases reported. earliest circa 1910.
- Contaminated product or bypnduct exported to U.S.
- Contaminated V slag exported to Austna; detected in Italy.
was discovered when the furnace dust was sent to a U.S. ore or as a decay product from uranium in the ore. The recycling facility to recover some of the other metals total activity concentration was about 150 Bq g-' (4 nCi (e.g., nickel) present in the dust, and radioactivity was g detected in the dust. In 1997, a health physicist conduct- (0.6'),mrad resultiny)in h' to 30 pGy contact radiation levels of h-' (3 mrad h-'), principally 6 Gy h" ing a routine contamination survey of vinyl lead aprons from the beta radiations emitted from those radioiso-in a hospital in Georgia, U.S., discovered that the lead topes. At least three lead vinyl apron manufacturers in itself was radioactive. The contaminant was identified as the U.S. were affected, and the lead aprons weie distrib-2'"Pb and its progeny,2'"Bi and 2'"Po (U.S. NRC 1997). uted in several countries. In addition to the contaminated The vinyl lead was made in the U.S. using lead from a lead vinyl, other lead products ranging from (fuel) lead-bismuth-tin slag imported from a tin smelter in additives to lead golf club weights were also contami-Brazil. It is believed that the 2t"Pb came either from a nated. These cases again illustrated the international radium source that was accidentally mixed with the tin aspect of the problem.
Radioactne matenah in rec)cied metals O1 o. La m w no 10 Yi wo 295 A 1995 melting of "'Co at a Czech steel mill first were gold, copper, and lead (2 cases each) and a single came to light as a result of informal discussions at the case involving zine. 1996 IRPA Congress. A Czech newspaper report quoted The database was greatly expanded by the sharing of an official of the Regional Centre of the State Office for data by the Steel Manufacturers Association (SMA) Nuclear Safety stating that "[tjhere have been five similar beginning in 1994 and inclusion of data for Canada cases in the Czech Republic in the last 6 years" but the (Table 2). Data are also provided by States who report authors have yet to confirm this.1 the usage of U.S. Department of Transportation exemp-1995 was the Grst year since 1986 that there was no tion for returning radioactively contaminated metal scrap report of a melting of a radioactive source in the U.S.; shipments and by other sources. Through 30 June 1997, however, meltings involving recycled aluminum oc- the database, which is maintained by the authors, con-curred in the U.S. in April,1996, and in January,1997, tained 2,357 reports of radioactive materials found in in the April 1996 incident, which involved recycled recycled metal scrap (Fig.1). Of these, 62'7c involved aluminum scrap possibly obtained from a military facil- NORM. Another 259 were unidentiGed as to the nature ity, thorium was found as the contaminate. The January of the contaminate. The remainder involved radium, 1997 incident involved *Am. which was found as a radioactive materials subject to the Atomic Energy Act, contaminate in the furnace, aluminum ingots, slags and and accelerator-produced radioactive materials. the area proximal to the furnace. Reports of radioactive materials in metal scrap There were no reports of the U.S. steel mills melting trended upwards through 1995 (Fig. 2). The reason for radioactive sources in 1995 or 1996. In 1997, a steel mill the drop in 1996 is not clear. It must be remembered that in Kentucky melted a '"Cs source. Also, in 1997, steel there are no requirements for reporting of these events. scrap contaminated with "'Co was detected and identi- Thus, the authors are dependent upon voluntary sharing fied by a steel mill in Pennsylvania. The contaminated of information by the metal recycling industry and scrap came from a nearby shovel manufacturer who information collected by the NRC and the State radiation stamped shovel blades from steel coils manufactured by control programs. Factors that are beyond the authors' a steel mill in Ohio. The "'Co levels in the steel were control, such as fear of litigation or financial risks, may about 300 Bq g" (8 pCi g-'). Contact gamma radiation affect the reporting rate. levels from the steel ranged from 0.1 to 0.6 pGy h~ ' (10 Scaled sources, devices containing sealed sources, to 60 prad h"). It is possible that over 25,000 metric or discrete sources such as alloys containing radioactive tons of steel may have been contaminated. The Ohio steel materials were reported found in metal scrap on 244 mill that made the steel coils uses nuclear gauges on its occasions in the U.S. and Canada (Table 3). Radium process line and has embedded "'Co sources in its blast sources accounted for 116 (489) of the reports. Sources furnace as refractory brick wear indicators. Following or devices containing radioactive materials subject to the discovery of the contaminated steel scrap, the NRC U.S. Atomic Energy Act of 1954, as amended, were the performed an inspection of the mill. This disclosed that most common-reported i 19 (499 ) times. Of these, the all of the licensed souices were accounted for. With most frequently found, by far, were '"Cs sources-respect to the "'Co refractory brick wear indicators, the reported on 52 (21'7<) occasions. This is not surprising erosion of such sources normally proceeds at a sery low since '"Cs is the most common radioactive source used rate resulting in levels in the iron (and steel made from the iron) that are not detectable except by using low-level radiation detection equipment. The Environmental Pro-tection Agency advised the shovel manufacturer that the Table 2. Recent metal scrap contamination events-U.S. and radiation levels did not constitute a health hazard." canada. Nonetheless, the manufacturer, who had not distributed year incidenis Ateinngs I any of the contaminated stampings, returned all of the g 3 l 3
"'Co-contaminated steel. The steel mill also monitors g4 2 i incoming scrap metal for radioactivity, but the monitors 85 m i
]
did not show any unusual radiation levels on the metals F6 14 n that were received by the mill for recycling. The source 2' [ { of the contamination as yet remains unexplained. 39 33 3 Worldwide 49 accidental meltings of radioactive 90 i33 i sources have been confirmed, including 28 in the U.S. 91 lis i The most frequently involved were ferrous metal fur- 92 158 4 naces with 35 meltings. Aluminum smelters were in- y $ 5 volved in 7 meltings. Other recycled metals affected 93 hs. In 96 335' l 97' 163 3 1"Radioacthe Steel 'on Ice
- for llalf Century." by J. Rosendorf.
15 February 1996 inglish translation nenpaper not specifiedi. " includes data from the Steci Alanufacturers Awocution membert
" letter from W. E. Helanger. EPA to M. Wherley. True Temper
- Canada.
Tooh. Inc., dated 17 April 1997. ' Through June 3n. l l
2V'. Ilealth Ph sics March 1998. Volume 74. Number 3 NaturaHy Occurring 'lable 3. Radioactise sources or desices found in U.S. and Unidentifled - 25% Radioactive Maternal Canadian ree)cled metal scrap-1983-June 1997. (NORM)- 62% T) pe No. Reporb W ) Maierials subject to the U.S. I19 (49) Atomic Energy Act. as amended i"cs 32 Other - <1% Th 23 U l7 Radium - 6% N Am 1J
"Co 6 "Kr 3 Atomic Energy Act
""Sr '
~
Material- 7% Radium i16 (4M) l Fig.1. Radioactive materials found in U.S. and Canadian recycled Other or unidenufied 9 (3) metal scrap. 1983-30 June 1997 (2.357 reports). Total 244 0 00) em , reinforcing rods (rebar) that were contaminated by "'Co 7,_ were used in construction of buildings. The Taiwanese Atomic Energy Council (AEC) has confirmed that the
,, soo- contaminated rebar was produced in late 1982 and early c 1983 as the result of meltings of multiple "'Co sources, j 5"~
the largest of which was estimated at 740 GBq (20 Ci) g go_ ( AEC 1994; Chen 1996). Wushou P. Chang, of the l j National Yang Ming University Medical School, re-l E soo- ported that 120 buildings constructed between 1982 and j 1984 have been confirmed by the Taiwanese Atomic 2x-Energy Council to contain " excessive amounts of Cobalt-
,,,_ 60" (Chang 1993; Chang and Kau 1993; Chang et al.
1997)." More than 5.000 members of the public may o- have been exposed. In one apartment house, which is es e4 es se er sa so so si s2 es e4 es es'er believed to be the most contaminated building. I14 mn, residents were exposed to radiation levels from 0.4 to 120 pSv h-' (0.N to 12 mrem h~'). The doses received Fig. 2. Metal scrap contamination esents reported in U.S. and by the residents over 9 y are estimated to be from 0.067 Canada. 1983-30 June 1997. to 1.2 Sv (6.7 to 120 rem). The Taiwanese Department of Health has begun efforts to follow up the radiation health effects on the residents of this apartment building. The in licensed measurement devices (e.g., presence, thick- U.S. National Institute for Occupational Safety and ness, or density of material) in the U.S. In 9 cases (39 ), }{eai,h and the National Yang Ming University Medical the radioactive material was not identified. School in Taiwan are collaborating in the development of The authors' database has been shared with the modeh for estimating the dose to inhabitants of the NRC. In 1996, at the request of NRC Commissioner contaminated buildings (Cardarelli et al.1997). Greta Joy Dieus, NRC staff began a review of the This case illustrates that significant radiological database by comparing it with the data contained in consequences can result from the melting of radioactive NRC's Nuclear Materials Esents Database (NMED). material that has become mixed with recycled rnetal Preliminary results of the comparison indicate that for scrap used to make new metal products that are widely reports of interest to the NRC, i.e., those invohing disseminated. It also shows the potential for long delays sources subject to the Atomic Energy Act, many of the to occur between the event and its subsequent discovery, events in the authors' database were also found in the exacerbating the public exposure problem. NMED, but this was not true in all cases.** Work will continue to sort out the differences. i TEXAS OVEREXPOSURE INCIDENT TAIWAN UPDATE in 1996 industrial radiography devices containing Additional estimated exposure information has re- "'Co and '"2Ir sources that were in storage in llouston, cently become available for un incident in Taiwan, w here Texas were stolen and sold as metal scrap'. During the multiple transters of the devices, one of the 'Co sources
** NRC Memorandum dated 19 December 1996. from E. Jordan. - - - - - - -
Director. Office of Analysis and Esaluation of Operating Data to J
' Additional information on this incident was provided by C hang Commiwioner Dicut to the authors during his sisit to the U.S. in 1997.
Radmactne matenals in recycled metah O L o. La m sw wn J G. Yt sso 297 containing 1.3 TBq (35.3 Ci) was dislodged from its Control, the annual meeting of the Conference of Radi-holder and fell to the ground near the office of a metal ation Control Program Directors, Inc.. in May 1996. A scrap handling firm. The stolen devices, including the concerted elfort was made to involve representatives all unshielded source, were recovered and arrests were potential stakeholders. At one workshop, held in January made. Workers and customers of the scrap yard and law 1996, in Washington, DC, representation included the enforcement officers w ho conducted investigations at the metal recycling industry, the steel industry, organized scrap yard were exposed to the source and may have labor, professional health and safety organizations, li-received doses up to 100 mSv (10 rem). One worker w ho censed users and source and device manufacturers and handled the source received an overdose to an extremity distributors. The report and recommendations of the (U.S. NRC 1996)." While no melting of a radioactive Working Group were conveyed to the Conunission on 2 source occurred in this incident, it illustrates the risks July 1996, and published in October 1996 (U.S. NRC resulting when radioactive sources enter the public do- 1996). main in an uncontrolled manner. The radioactivity in the The Working Group concluded that additional reg-stolen *lr source had decayed and presented no expo- ulatory action is warranted, and recommended that "l) sure ha/ard in this incident. NRC and Agreement States increase regulatory osersight for users of certain devices: 2) NRC and Agreement RECENT NRC ACTIONS S'"ICS I*P"*# P#""I'ICS "" P'IS""" I"'I"E dC*IC *; 3I NRC and Agreement States ensure proper disposal of. Of an estimated 2 million licensed devices contain. orphaned devices;" 4) NRC encourage States to imple-ing radioactive material in the United States, about ment similar oversight programs for users of Naturally-one-quarter are used under a specific license and three. Occurring or Accelerator-Produced Material; and 5) quarters are used under a general license, primarily the NRC encourage non-licensed stakeholders to take appro-general license in 10 CFR 31.5 or equivalent Agreement priate actions, such as instituting programs for material State regulations (U.S. NRC 1996"). NRC requires identihcatmn., reporting to it of initial transfers of generally licensed in Nosember 1996, the Commi.,sion directed the devices to general licensees but thereafter its oversight staff to move expeditiously to develop plans to address (and that of many but not all Agreement States) of the Working Group recommendations." To maintain general licensees is minimal-they are not subject to compatibility, the Agreement States are also expected to routine inspections, license renewals, or other periodic develop similar plans to improve the control and ac-regulatory contacts. In June 1995, the Nuclear Regula. countability of licen ed devices, if these States have not tory Conunission approved a staff recommendation to already done so. The Working Group noted that several for'm a joint Agreement State-NRC Working Group to of the Agreement States have previously implemented evaluate the problem oflicensees not maintaining control programs that addressed several of its recommendations. and accountability of devices containing radioactive material and to propose solutions.* The Working Group TRADE AND INTERNATIONAL, ACTIVITIES held 8 public meetings including a workshop and partic-ipated in the 28th National Conference on Radiation Americtm Metal MarAct, a weekly trade magazine for the metal recycling industry, has frequently published
" See appenda 11 of the reference for details of this esent. The news reports on the problem of radioactive materials in appendis includes prehminary dose estimates of up to $30 mSv (53 recycled metals. On 7 December 1995, its supplement, remt Subsequent to publication of the reference, Texas official * " Mini-Mill Steel." featured a two page report, "It's war informed NRC staff that cytopenetic studies indicated that none of the rsons exposed to the source receised doses in exceu of 100 mSv (10 on radioactivity," (Worden 1995L Scrap, the journal of i of Scrap Recycling Industries (ISRIL pub-
" This report. NUREG-1551, prm ides an estimate of 1.5 mdlion lished an article, "The Radioactive Scrap Threat," w hich generally beensed desices in the U.Sa see p. 25. In 1989, a mail sun ey reported that " Radioactive sources in the scrap stream are conducted by a NRC contractor resuhed in a report that oser 227,(XX) sealed sources a .d desices held under NRC specific licenses see a serious problem for recyclers and consumers. '
Table I in "An insentory of NRC licensed scaled sources and desices --- final report. December 31,1991" prepared by Oak Ridge Associated 'N " Orphaned des ices" are des ices containing radioactis e material Unis ersitiet After adjustments for the suney response rate and to that are found in the pubhc domain ttypically by metal recyclers) that , account for sources and desices also held under Agreement State cannot be traced to the owner or heensee because labeh or other specific licemes the authors estimate another K(MMMM) sources and identifying markings are no longer present, legible, or accewible. h desices are used in the U.S. under a specine beense. The majonty of may ahn be impossible to determine ounership of the desices due to seated sources are contained in desicet Gisen the uncertainties of closure of the manufacturing or distubuting beensee. In such cases the these estimates the authors. after addmp them together, rounded of f finder is usually responsible for securing the desice for safety and for the total to a smgle significant figure leathng to the estimate of 2 disposing ofit. The resulting costs can be substantial, especiaHy if the million desices held under general and specifie licenses in the U.S. desice must be dnposed of as low lesel radioactise waste. The Note that this figure does not include radium and accelerator produced Workmg Group concluded that such cmts can act as a dnincentne for radmactise sources nor sources powessed by the U.S. Department of findmg and reportmg of such source and. therefore, recommended that Enery) . ways be found to remose or mitigate these dnincentises.
" NRC Staff Requirements Memorandum ICOMGD-944X)3) " Staff Requirements Memorandum dated 31 December 1996.
dated 18 October 1994, from J. Iloyle Secretary to J. Taylor, from J. C. Iloyle, Secretary to J. M. Taylor, lhecutise Ihrector for lhecutise Director for Operations. Operationt
v 298 Ilealth Phpics March 1998. Volume 74. Number 3 (Kizer 1996). ISRI has also published a 40-page booklet, including metals."' The metal recycling industries are
" Radioactivity in the Scrap Recycling Process: Recom- concerned about these developments (Bechak 1997; mended Practice and Procedure" (ISRI 1993). A com- Nieves et al.1995). The radiation from radioactively panion training videotape is available from ISRI in both contaminated metals released for unrestricted use could English and Spanish versions.*** A Spanish language confound their radiation monitoring of metal scrap to version of the Recommended Practice and Procedure is detect radioactive sources mixed in the metal scrap that being prepared. could contaminate their plants and expose their workers.
Metal scrap processors and consumers who realize Thus, regulatory agencies that propose standards for that they need to perform radiation monitoring of their unrestricted recycling of radioactively contaminated met-metal scrap are faced with a bewildering assortment of als may need to factor in this aspect as well as the choices of radiation detection equipment and costs. To radiological health risks resulting from the radioactive provide some insights, in 1996, the SMA sponsored field contamination of the metal. tests of commercially available radiation detection equip- The metal recycling industry is already faced with ment used to detect radioactive materials in incoming this problem as a result of their frequent discovery (62%) metal scrap shipments. A report of the results of the field of NORM contaminated scrap metal. Arguably. NORM tests became available in 1997."' contaminated metal scrap has been handled and con-The 1996 International Radiation Protection Asso- sumed by the metal scrap industry for years but went ciation Congress included a poster paper by the authors unnoticed until the widespread use of radiation monitors (Lubenau and Yusko 1996). During the Congress, a that were intended to detect radioactive sources and panel addressed the subject of radiation source safety and devices. Without Federal or other guidance, mills have reviewed the history of mishaps and overexposures been reluctant to melt metal scrap that has been identified involving radioactive sources. The panel concluded that as contaminated with NORM. As a result, when such there is a need to provide greater assurance that large metal scrap is found, it is usually returned to the shipper radioactive sources are appropriately controlled. Several or,if accepted by the recipient,is segregated and stored. vendors of radiation detection equipment at the Congress advertised systems specifically targeted for metal recy. CONCLUSIONS tiers. evidence that radiation monitoring of recycled metal has become a widely recognized market niche for The problem of radioactive contamination of recy-vendors of radiation detection systems and consultants. cled metals is the most common and most visible manifestation of a larger problem-inadequate control, insufficient accountability, and improper disposal of IMPLICATIONS OF THE RECYCLING OF radioactive materials. As a result, radioactive sources are RADIOACTIVELY CONTAMINATED METALS entering the public domain in an uncontrolled manner thus creating a risk of unnecessary and sometimes la 1992, the IAEA . issued recommended standards excessive exposure of workers and the public. The true for the recycling of radioactively contaminated matenals scope of the problem remains to be described because miended for "f,ree release," i.e., without contmls or some entities are reluctant to report events because of restrictions on subsequent use of the material. These legal considerations, financial risks, or concern over enteria are based upon consideration of the radiological negative reaction or perception. This is, in the authors' health nsks resultmg from the subsequent use of such op nion, unfortunate and misguided. More information, matenals. In the U.S., entena exist that can be applied t not less, is needed if the problem of uncontrolled or matenals havmg surface contammation that are miended mproperly disposed radioactive materials is to receive for unrestricted use (U.S. NRC 1974) but no standards appropriate attention by national and international radi-exist for volumetric contamination. The U.S. Department ation safety bodies. Means must be found to encourage of Energy, faced with a large mventory of radioactively the release and dissemination of infonnation about this I contammated metals, has imtiated a program for recy- problem to help increase awareness in the industry and to cling
- radioactively contaminated metals meludm, g carbonstimulate protective measures and regulatory responses.
steel. ** The U.S. Environmental Protection Agency and in the U.S., the NRC is considering taking steps to the NRC have announced plans to develop criteria to improve regulatory assurance that licensees maintain permit recychng of radioactively contaminated materials appropriate control and accounting of devices containing . radioactive material. However, even if additional regu-latory actions are approved, it will take time for these
'" Institute of Scrap Recycling industries. Inc.,1325 G Street measures to take effect. Further, while improvements in NW. Suite HXR Washington. DC 2on05-31M s
'" " Data analysis. Steel Manufacturers Association. test of scrap licensee and regulator performance with respect to con-monitoring systems, Koppel Steel Koppel. PA. September, 23- trol and accounting will be expected as a resuh, given the Oct her 41996," Health Physics Associates. Ir.c.10n5 Old Route 22. huge population of radioactive devices in the U.S., total irnhartmile. PA 19534 (2 June 1997L
*" DOE Memorandum from Alvia L. Alm. Assistant Secretary for Ensironmental Management to Distribution. subject: Policy on
- NRC Memorandum 40m L.J. CaHan to the Commissioners recycling radioactively contaminated carbon steel dated 20 September dated 5 June 1997; subject: 5tatus of the NRC recycle and reuse staff i 1996. action plan. SECY-97-Il9. f l
Radmactne materiah m recycled meiah O L O. l.t m su no J. G. Yiwo 299 elimination of all incidents of loss of control of radioac- sure: Causes and challenges. Heahh Phys. 73:465-472; tise devices is not realistic. Constant awareness and 1997. vigilance by the metal scrap recycling industry must Chang, W. P.; Kau. J. Taiwan: Expo ure to high doses of remain a part of that industry. radiation. Lancet 341:750; 1993. Chen Y. Lessons learned from the recovery operations in the cobalt-60 contaminated rebar accident. Taiwan: Atomic Energy Council Department of Nuclear Technology; 29 Admmledgmenn-Thn paper does not represent agreed upon staff October 1996. positions of the NRC or the Penns)hania ikpartment of Ensironmental Protection, nor base these agencies approsed the techmcal content. International Atomic Energy Agency. Application of exemp-The authors appreciate the conimuing cooperation and assi tance of tion principles to the recycle and reuse of materials from numerous State. NRC and other Federal staf fs, foreign posernment nuclear facih. .tles. Vienna: IAEA Safety Series no. Ill-P-agencies, the ualh of the Institute for Scrap Recychng indusines. Inc.. the bI; I992-Steel Manufacturers Association. the American iron and Steel Institute, the Institute for Scrap Recycling Industries. Inc. Radioactivity in Conference of Radiation Control Prograrn Directors Inc. and representa- the scrap recycling process-Recommended policy and pro-thes of industries insobed in the recychng of metal scrap. cedure. Washington, DC: ISRI Report; 12 January 1993. Notes-tLS Gosernment reports and pubhcanons cited herein are asailable for purchase from the Nanonal Technical Information Senice. Ki/er, K. Radioactise scrap threat. Serap 53:175-186; 1996. Springfield, yA 22161. Copies of referenced SMA. NRC DOE and EPA Lubenau,1 W. i.usko, J. G.; Radioactis e materiah .in rec)cled documents are asailable for inspection and copying for a fee in the NRC metals. llealth Phys. 68:440-451; 1995. Pubhc Document Room. l.ou er Ecsel 11.1.-6). 212n 1. Street Nw. Lubenau, J. O.; Yusko, J. G. Radioactive contamination of Washmpton DC. recycled metals.1996 international Congress on Radiation Readen are encouraged to contact the authors to share informahon on Protection Proceedings 3:291-293; 1996. events insoh mg radioacthe contamination of recycled metak. Copies of Nieves. L. A.; Chen. S. Y., Kohout. E. J.; Nabelssi, R W., the authors' database can be obtained by uritmg to J. Yusko or e-mmhng him at >uskojameda a1. der state pa us. Tilbroi R. % Wihom S. E. Esaluation of radioactise g ;; g gggg$ 1995. REFERENCES U.S. Nuclear Regulatory Commission. Termination of operat-ing licenses for nuclear reactors. Washington, DC: NRC Atomic Energy Council. Executise Yuan, Republic of China. Regulatory Guide 1.86; 1974. Contaminated rebars, incident report. Taiwan: AEC-083 - 01-201, 021013830091; I994. U.S. Nuclear Regulatory Commission. Final report of the NRC-Agreement State working group to evaluate control liechak, C. Who s responsible for radioactise scrap? Iron Age /New Steel 13:7; 1997. and accountability of licensed desices. Washington, DC: NUREG-1551; 1996. Cardarelli,11. J.; Elliott. L.; llornung, R.; Chang % Proposed model for estimating dose to inhabitants Co of l. P.Nuclear Regulatory Commission. Contaminated lead U.S. contaminated buildings. flealth Phys. 72:351-360; 1997. oduch Washington DC: NRC Information Notice 97-Chang. W. P. Spread of Taiwan's radiation panic. Lancet . 'g g 997' , 342:1544; 1993. orden. E. It s war on radioactivity. American Metal Market: Chang, W. P.; Chan, C. C.: Wang, J. D. "Co contamination in bupplemeni) 7 December,1995. ree>cled steel resulting in elevated cisilian radiation espo. EE 1 l}}