Bayesian Statistics Applied to Reliability Analysis ...

1 9/23/12 1 Bayesian Statistics Applied to Reliability Analysis and Prediction By Allan T. Mense, , PE, CRE, Principal Engineering Fellow, Raytheon Missile Systems, Tucson, AZ 1. Introductory Remarks. Statistics has always been a subject that has baffled many people both technical and non technical. Its basis goes back to the mid 18th century and the Analysis of games of chance. Statistics is the application of probability and probability theory can be traced to the ancient Greeks but it was most notably developed in the mid 17th century by the French mathematicians Fermat, Laplace, and others. Rev. Sir Thomas Bayes (born London 1701, died 1761, see drawing below) had his works that includes the Theorem named after him read into the British Royal Society proceedings (posthumously) by a colleague in 1763.

2 I have actually seen the original publication! For years and even in the present day the Statistics community seems to have a schism between the so-called objectivists or frequentists and their so-called classical interpretation of probability and the Bayesians who have a broader interpretation of probability. From a Reliability point of view classical calculations can be thought of as a subset of Bayesian calculations. You do not have to give up the classical answers but you will have to give up the classical interpretation of the results! Discussion of the logical consistencies and inconsistencies of the two statistical points of view would lead us too far afield [5] . However my personal observations indicate that in the battle over which techniques to apply to problems, the Bayesian have won the war but classical techniques are still widely used, easy to implement and are very useful.

3 We will use both but the purpose of this note is to explain Bayesian techniques Applied to Reliability . References. The two major texts in this area are Bayesian Reliability Analysis , by Martz & Waller [2] which is out of print and more recently Bayesian Reliability , by Hamada, Wilson, Reese and Martz [3]. It is worth noting that much of this early work at Los Alamos was done on weapon system Reliability and all the above authors work or have worked at the Los Alamos National Lab [1]. Allyson Wilson headed the Bayesian Reliability group at LANL and Christine Anderson-Cook now heads that group. They are arguably among the leading authorities in Bayesian Reliability in the world. I will borrow freely from both these texts and from notes I have from their lectures. There are also chapters covering Bayesian 9/23/12 2 methods in traditional Reliability texts Statistical Methods for Reliability Data, Chapter 14, by Meeker and Escobar [4].

4 This point paper covers Bayesian Reliability theory and Markov Chain Monte Carlo (MCMC) solution methods. The NIST web site also covers Bayesian Reliability . Specifically covers What models and assumptions are typically made when Bayesian methods are used for Reliability evaluation? Philosophy. The first and foremost point to recognize is that Reliability has uncertainty and therefore should not be thought of as a single fixed number whose unknown value we are trying to estimate. Reliability having this uncertainty requires us to treat Reliability as a random variable and therefore discuss it using probability distributions, f(R), and the language of Statistics how likely is it that Reliability of a system or component will have some value greater than some given number (typically a Reliability specification). We will see that specifying some desired Reliability value is not sufficient but requires that we also specify some level of confidence that Reliability is greater than (or less then) the value desired.

5 This will become clear when Reliability distributions are defined and calculated. For those needing a refresher on Statistics I recommend Introduction to Engineering Statistics , by Doug Montgomery et al and any edition of this text will do just fine. The fundamental concepts you will need are 1) probability density functions, pdf, written as f(x|a,b,c) where the letters a,b,c refer to pieces of information, called parameters, that are presumed known, knowable or can be estimated , 2) cumulative distribution function (CDF), written F(x|a,b,c), which is the accumulated probability of the random variable X from the minimum allowable value of X (typically zero or - ) up to X=x, and 3) the concept of a likelihood function which in practice is the product of pdf s and CDF s evaluated in terms of all the available data.(See Appendix G ). The use of a likelihood function while familiar to every statistician is not used by everyday Reliability engineers!

6 The concept was proposed by Sir Ronald Fisher back in the early 1900 s and is very useful. For those not familiar with the traditional frequentist method for establishing a Reliability estimate and its confidence interval, Appendix C has been provided. Why Bayesian Bayesian methods make use of well known, statistically accurate, and logically sensible techniques to combine different types of data, test modes, and flight phases. Bayesian results include all possible usable information base on data and expert opinion. Results apply to any missile selected and not just for average sets of missiles . Bayesian methods are widely accepted and have a long track record: FAA/USAF in estimating probability of success of launch vehicles Delphi Automotive for new fuel injection systems Science-based Stockpile Stewardship program at LANL for nuclear warheads Army for estimating Reliability of new anti-aircraft systems FDA for approval of new medical devices and pharmaceuticals 9/23/12 3 It is also worthy of note that Bayesian Reliability has been actively pursued for at least 30 to 40 years and the Los Alamos National Lab (LANL) developed the techniques to predict the Reliability of missiles as well as the nation s nuclear stockpile back in the 1980 s and The reasons for using Bayesian can be summarized and are outlined in the chart shown on the previous page.

7 Before proceeding in detail there is a simple picture to keep in mind that explains how Bayesian Reliability analyses works. One starts by (1) producing the blue curve (prior distribution) based upon previous information deemed useful for predicting the probability of successfully operating units prior to taking data on the system or component of interest, then (2) folding in recent test data as is represented by the red distribution (likelihood function) and finally (3) producing, using some rather sophisticated math in the general case, a curve such as shown in green in Figure 4. The green curve represents the posterior distribution of a unit s Reliability given the most recent data. From this green curve we can in principle, calculate everything that is needed. Figure 7. The Prior, Likelihood and Posterior, superimposed. One postulates from previous information a prior distribution shown in blue.

8 The tests are performed and put into a likelihood function shown graphically in red. The result (in green) is the answer. It is the posterior Reliability distribution. Note that the posterior distribution (green) is more peaked and narrower than the (red) likelihood curve which is indicative of having prior information on the Reliability . The likelihood function (red) by itself would be the distribution found from classical or frequentist Analysis from a prior that is uniform. 9 Posterior DistributioncombiningPriorand Data Bluecurve is our priordistribution Redcurve is distribution assuming only information is 4 successes in 5 tests Greencurve is the Bayesian estimate adding the 4 out of 5 to the prior Estimate is betweenevidence (data) and prior Distribution is tighter than prior ordata only narrower confidence ReliabilityProbability Density FunctionPosterior distribution of System Reliabilityafter 4 successes in 5 testsInitial distribution of System Reliability assuming the most probable value is of System Reliability using 4 successes in 5 tests only 9/23/12 4 Finding the (green) posterior distribution for real situations is mathematically complex, but the essence of what is done is as simple as the graphical display shown above.

9 For those caring to delve further in the details, the following sections have been provided. Table of Contents. Introductory remarks References Philosophy Overview Basic Principles Bayes Theorem: Prior, Likelihood, Posterior Bayes Theorem Allied to Pass/Fail Reliability General Bayesian Approach to Reliability General Procedure for Bayesian Analysis and Updating Selecting a Prior Likelihood Function Generating System Level Reliability Estimate Summary Time Dependent Reliability Calculations Using a Weibull Distribution Poisson Counting Appendices 2. Overview It makes a great deal of practical sense to use all the information available, old and/or new, objective or subjective, when making decisions under uncertainty which is exactly the situation one has with many systems in the field. This is especially true when the consequences of the decisions can have a significant impact, financial or otherwise.

10 Most of us make everyday personal decisions this way, using an intuitive process based on our experience and subjective judgments.[6] Using language from the NIST web site we note that so-called classical or frequentist statistical Analysis , seeks objectivity by generally restricting the information used in an Analysis to that obtained from a current set of clearly relevant data. Prior knowledge is not used except to suggest the choice of a particular population model to "fit" to the data, and this choice is later checked against the data for reasonableness. What is wrong with this approach after all it has used successfully for many years? The answer lies in the desire to take into account previous information particularly if we have some information from flight tests and some from ground tests and we want to somehow combine this information to predict future probabilities of success in operational scenarios.