Unit 14: Introduction to the Use of Bayesian Methods for Reliability Data. Ramón V. León

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1 Unit 14: Introduction to the Use of Bayesian Methods for Reliability Data Ramón V. León Notes largely based on Statistical Methods for Reliability Data by W.Q. Meeker and L. A. Escobar, Wiley, 1998 and on their class notes. 9/26/2004 Stat 567: Unit 14 - Ramón León 1 Unit 14 Objectives Describe the use of Bayesian statistical methods to combine prior information with data to make inferences Explain the relationship between Bayesian methods and likelihood methods used in earlier chapters Discuss sources of prior information Describe useful computing methods for Bayesian methods Illustrate Bayesian methods for estimating reliability Illustrate Bayesian methods for prediction Compare Bayesian and likelihood methods under different assumptions about prior information Explain the dangers of using wishful thinking or expectations as prior information 9/26/2004 Stat 567: Unit 14 - Ramón León 2

2 Introduction Bayes methods augment likelihood with prior information A probability distribution is used to describe our prior beliefs about a parameter or set of parameters Sources of prior information Subjective Bayes: prior information subjective Empirical Bayes: prior information from past data Bayesian methods are closely related to likelihood methods 9/26/2004 Stat 567: Unit 14 - Ramón León 3 Bayes Method for Inference 9/26/2004 Stat 567: Unit 14 - Ramón León 4

3 ( B) i ( θ x) 1 2 Bayes Theorem ( ) ( ) P B A P A PAB ( ) = P B A P A P A ( ) ( ) + P( B A) P( A) n i= 1 ( ) ( ) P B A P A ( ) ( ) where A, A,..., A is a partition of the sample space f = = n ( θ) ( θ) ( ) ( ) i P B A P A f x f f x θ f θ dθ i i i 9/26/2004 Stat 567: Unit 14 - Ramón León 5 Updating Prior Information Using Bayes Theorem 9/26/2004 Stat 567: Unit 14 - Ramón León 6

4 Some Comments on Posterior Distributions 9/26/2004 Stat 567: Unit 14 - Ramón León 7 Differences Between Bayesian and Frequentist Inference Nuisance parameters Bayes method use marginals Large-sample likelihood theory suggest maximization There are not important differences in large samples Interpretation Bayes methods justified in terms of probabilities Frequentist methods justified on repeated sampling and asymptotic theory 9/26/2004 Stat 567: Unit 14 - Ramón León 8

5 Sources of Prior Information Informative Past data Expert knowledge Non-informative (or approximately noninformative) Uniform over range of parameter values (or function of parameter) Other vague or diffuse priors 9/26/2004 Stat 567: Unit 14 - Ramón León 9 Proper Prior Distributions 9/26/2004 Stat 567: Unit 14 - Ramón León 10

6 Improper Prior Distributions 9/26/2004 Stat 567: Unit 14 - Ramón León 11 Effect of Using Vague (or Diffuse) Prior Distributions 9/26/2004 Stat 567: Unit 14 - Ramón León 12

7 Eliciting or Specifying a Prior Distribution The elicitation of a meaningful joint prior distribution for vector parameters may be difficult The marginals may not completely determine the joint distribution Difficult to express/elicit dependences among parameters through a joint distribution. The standard parameterization may not have practical meaning General approach: choose an appropriate parameterization in which the priors for the parameters are approximately independent 9/26/2004 Stat 567: Unit 14 - Ramón León 13 Expert Opinion and Eliciting Prior Information Identify parameters that, from past experience (or data), can be specified approximately independently (e.g. for high reliability applications a small quantile and the Weibull shape parameter). Determine for which parameters there is useful informative prior information For parameters for which there is no useful information, determine the form and range of the vague prior (e.g., uniform over a wide interval) For parameters for which there is useful informative prior information, specify the form and range of the distribution (e.g., lognormal with 99.7% content between two specified points) 9/26/2004 Stat 567: Unit 14 - Ramón León 14

8 Example of Eliciting Prior Information: Bearing-Cage Time to Fracture Distribution 9/26/2004 Stat 567: Unit 14 - Ramón León 15 Example of Eliciting Prior Information: Bearing- Cage Fracture Field Data (Continued) 9/26/2004 Stat 567: Unit 14 - Ramón León 16

9 Example of Eliciting Prior Information: Bearing- Cage Fracture Field Data (Continued) 9/26/2004 Stat 567: Unit 14 - Ramón León 17 9/26/2004 Stat 567: Unit 14 - Ramón León 18

10 9/26/2004 Stat 567: Unit 14 - Ramón León 19 Joint Lognormal-Uniform Prior Distributions PX ( σ ) = P(logX log σ ) 9/26/2004 Stat 567: Unit 14 - Ramón León 20

11 9/26/2004 Stat 567: Unit 14 - Ramón León 21 9/26/2004 Stat 567: Unit 14 - Ramón León 22

12 Methods to Compute the Posterior 9/26/2004 Stat 567: Unit 14 - Ramón León 23 Computing the Posterior Using Simulation 9/26/2004 Stat 567: Unit 14 - Ramón León 24

13 P Proof of Simulation Algorithm ( θ θ θ ) 0 P accepted = 0 ( θ θ0, θ accepted) P ( θ accepted) ( θ θ0 U R θ ) PU ( R( θ )) P, ( ) = = ( ) ( ) 9/26/2004 Stat 567: Unit 14 - Ramón León 25 0 PU R( θ ) f( θ) dθ PU R( θ ) f( θ) dθ θ0 R( θ) f( θ) dθ θ 0 R( θ) f( θ) = = dθ R( θ) f( θ) dθ R( θ) f( θ) dθ = θ f( θ x) dθ θ 9/26/2004 Stat 567: Unit 14 - Ramón León 26

14 9/26/2004 Stat 567: Unit 14 - Ramón León 27 Sampling from the Prior log t = log a + Ulog b p 1 1 9/26/2004 Stat 567: Unit 14 - Ramón León 28

15 9/26/2004 Stat 567: Unit 14 - Ramón León 29 9/26/2004 Stat 567: Unit 14 - Ramón León 30

16 9/26/2004 Stat 567: Unit 14 - Ramón León 31 Comment on Computing Posterior Using Resampling 9/26/2004 Stat 567: Unit 14 - Ramón León 32

17 Posterior and Marginal Posterior Distributions for the Model Parameters 9/26/2004 Stat 567: Unit 14 - Ramón León 33 Posterior and Marginal Posterior Distributions for the Functions of Model Parameters 9/26/2004 Stat 567: Unit 14 - Ramón León 34

18 9/26/2004 Stat 567: Unit 14 - Ramón León 35 Simulated Marginal Posterior Distribution for F(2000) and F(5000) 9/26/2004 Stat 567: Unit 14 - Ramón León 36

19 Bayes Point Estimation ( ĝ ( θ) g( θ) ) 2 f ( θ DATA) 9/26/2004 Stat 567: Unit 14 - Ramón León 37 One-Sided Bayes Confidence Bounds 9/26/2004 Stat 567: Unit 14 - Ramón León 38

20 Two-Sided Bayes Confidence Intervals bounds 9/26/2004 Stat 567: Unit 14 - Ramón León 39 Bayesian Joint Confidence Regions 9/26/2004 Stat 567: Unit 14 - Ramón León 40

21 Bayes Versus Likelihood 9/26/2004 Stat 567: Unit 14 - Ramón León 41 Prediction of Future Events DATA 9/26/2004 Stat 567: Unit 14 - Ramón León 42

22 Approximating Predictive Distributions 9/26/2004 Stat 567: Unit 14 - Ramón León 43 Location-Scale Based Prediction Problems 9/26/2004 Stat 567: Unit 14 - Ramón León 44

23 Predicting a New Observation 9/26/2004 Stat 567: Unit 14 - Ramón León 45 Predictive Density and Prediction Intervals for a Future Observation from the Bearing Cage Population 9/26/2004 Stat 567: Unit 14 - Ramón León 46

24 Caution on the Use of Prior Information In many applications, engineers really have useful, indisputable prior information. In such cases, the information should be integrated into the analysis We must beware of the use of wishful thinking as prior information. The potential for generating serious misleading conclusions is high. As with other inferential methods, when using Bayesian methods, it is important to do sensitivity analysis with respect to uncertain inputs to ones model (including the inputted prior information 9/26/2004 Stat 567: Unit 14 - Ramón León 47 SPLIDA Bayes Analysis 9/26/2004 Stat 567: Unit 14 - Ramón León 48

25 SPLIDA Bayes Analysis: Bearing Cage Data 9/26/2004 Stat 567: Unit 14 - Ramón León 49 9/26/2004 Stat 567: Unit 14 - Ramón León 50

26 9/26/2004 Stat 567: Unit 14 - Ramón León 51 Weibull Model Prior Distribution for BearingA data beta quantile beta log(0.01 quantile) log100 = 4.6 log 5000 = /26/2004 Stat 567: Unit 14 - Ramón León 52

27 9/26/2004 Stat 567: Unit 14 - Ramón León 53 Weibull Model Prior Distribution for Bearing Cage Data beta Sun Sep 29 18:24: quantile 9/26/2004 Stat 567: Unit 14 - Ramón León 54

28 9/26/2004 Stat 567: Unit 14 - Ramón León 55 Weibull Model Posterior Distribution for Bearing Cage Data beta quantile beta log(0.01 quantile) 9/26/2004 Stat 567: Unit 14 - Ramón León 56

29 SPLIDA Posterior of B10 9/26/2004 Stat 567: Unit 14 - Ramón León 57 Weibull Model Posterior Distribution for Bearing Cage Data f(t_0.1 DATA) Sun Sep 29 18:12: t_0.1 The mean of the posterior distribution of t_0.1 is: 2863 The 95% Bayesian credibility interval is: [2011, 4361] 9/26/2004 Stat 567: Unit 14 - Ramón León 58

30 Weibull Model Posterior Distribution for Bearing Cage Data f(beta DATA) beta Sun Sep 29 18:12: The mean of the posterior distribution of beta is: The 95% Bayesian credibility interval is: [2.184, 3.625] 9/26/2004 Stat 567: Unit 14 - Ramón León 59 9/26/2004 Stat 567: Unit 14 - Ramón León 60

31 Weibull Model Posterior Distribution for Bearing Cage Data beta Sun Sep 29 18:43: quantile 9/26/2004 Stat 567: Unit 14 - Ramón León 61 9/26/2004 Stat 567: Unit 14 - Ramón León 62

32 Weibull Model Posterior Distribution for Bearing Cage Data f(f(5000 DATA) F(5000) Sun Sep 29 20:12: The mean of the posterior distribution of F(5000) is: The 95% Bayesian credibility interval is: [0.1355, ] 9/26/2004 Stat 567: Unit 14 - Ramón León 63