Unit 14 Objectives - University of Tennesseeweb.utk.edu/~leon/rel/Fall04pdfs/567Unit14Handout.pdf2 9/26/2004 Stat 567: Unit 14 - Ramón León 3 Introduction • Bayes methods augment

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9/26/2004 Stat 567: Unit 14 - Ramón León 1

Unit 14: Introduction to the Use of Bayesian Methods for Reliability

Data

Notes largely based on “Statistical Methods for Reliability Data”

by W.Q. Meeker and L. A. Escobar, Wiley, 1998 and on their class notes.

Ramón V. León


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

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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


Bayes Method for Inference

3


Bayes Theorem( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )( ) ( )

1

1 2

|( | )

| |

||

|

where , ,..., is a partition of the sample space|

||

i ii n

i ii

n

P B A P AP A B

P B A P A P B A P A

P B A P AP A B

P B A P A

A A Af x f

f xf x f d

θ θθ

θ θ θ

=

=+

=

=

∑

∫


Updating Prior Information Using Bayes Theorem

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Some Comments on Posterior Distributions


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

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Sources of Prior Information

• Informative– Past data– Expert knowledge

• Non-informative (or approximately non-informative)– Uniform over range of parameter values (or

function of parameter)– Other vague or diffuse priors


Proper Prior Distributions

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Improper Prior Distributions


Effect of Using Vague (or Diffuse) Prior Distributions

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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


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)

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Example of Eliciting Prior Information: Bearing-Cage Time to Fracture Distribution


Example of Eliciting Prior Information: Bearing-Cage Fracture Field Data (Continued)

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Example of Eliciting Prior Information: Bearing-Cage Fracture Field Data (Continued)


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Joint Lognormal-Uniform Prior Distributions

( ) (log log )P X P Xσ σ≤ = ≤

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Methods to Compute the Posterior


Computing the Posterior Using Simulation

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Proof of Simulation Algorithm

( ) ( )( )

( )( )

( )

( )

0

0

0

0

00

0

P , accepted| accepted =

accepted

( ) ( )P , ( )

( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) ( )

( | )

PP

P U R f dU R

P U RP U R f d

R f dR f d

R f d R f d

f x d

θ

θ

θ

θ

θ θ θθ θ θ

θ

θ θ θθ θ θ

θθ θ θ

θ θ θθ θ θ

θ θ θ θ θ θ

θ θ

−∞∞

−∞

−∞∞ ∞

−∞

−∞ −∞

−∞

≤≤

≤≤ ≤

= =≤

≤

= =

=

∫

∫

∫∫

∫ ∫

∫


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Sampling from the Prior

1 1log log logpt a U b= +

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Comment on Computing Posterior Using Resampling

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Posterior and Marginal Posterior Distributions for the Model Parameters


Posterior and Marginal Posterior Distributions for the Functions of Model Parameters

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Simulated Marginal Posterior Distribution for F(2000) and F(5000)

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Bayes Point Estimation

( ) ( )( ) ( )2g g f DATAθ θ θ−∫


One-Sided Bayes Confidence Bounds

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Two-Sided Bayes Confidence Intervals

bounds


Bayesian Joint Confidence Regions

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Bayes Versus Likelihood


Prediction of Future Events

DATA

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Approximating Predictive Distributions


Location-Scale Based Prediction Problems

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Predicting a New Observation


Predictive Density and Prediction Intervals for a Future Observation from the Bearing Cage Population

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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 thinkingas 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


SPLIDA Bayes Analysis

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SPLIDA Bayes Analysis:Bearing Cage Data


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0.01 quantile 100 500 2000

123456

beta

5 6 7 8log(0.01 quantile)

2 3 4 5 6beta

Weibull Model Prior Distribution for BearingA data

log100 4.6log5000 8.51

==

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0.01 quantile 100 200 500 1000 2000 5000

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

beta

Sun Sep 29 18:24:50 2002

0.010.1 0.5 0.9

Weibull Model Prior Distribution for Bearing Cage Data

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0.01 quantile 500 1000 2000 4000

1.5

2.5

3.5

4.5

beta

6.5 7.0 7.5 8.0 8.5log(0.01 quantile)

1.5 2.5 3.5 4.5beta

Weibull Model Posterior Distribution for Bearing Cage Data

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SPLIDA Posterior of B10


t_0.11500 2500 3500 4500 5500

f(t_0

.1 |

DA

TA)

Sun Sep 29 18:12:26 2002


The mean of the posterior distribution of t_0.1 is: 2863The 95% Bayesian credibility interval is: [2011, 4361]

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beta2.0 2.5 3.0 3.5 4.0

f(bet

a | D

ATA)

Sun Sep 29 18:12:26 2002


The mean of the posterior distribution of beta is: 2.842 The 95% Bayesian credibility interval is: [2.184, 3.625]


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0.1 quantile 100 500 2000 5000 20000

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

beta

Sun Sep 29 18:43:33 2002

0.010.10.50.9



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F(5000)0.0 0.2 0.4 0.6 0.8 1.0

f(F(5

000

| DA

TA)

Sun Sep 29 20:12:46 2002


The mean of the posterior distribution of F(5000) is: 0.4589 The 95% Bayesian credibility interval is: [0.1355, 0.9071]

Documents

Unit 14 Objectives - University of Tennesseeweb.utk.edu/~leon/rel/Fall04pdfs/567Unit14Handout.pdf2 9/26/2004 Stat 567: Unit 14 - Ramón León 3 Introduction • Bayes methods augment