Bayesian Analysis by Markov Chain Monte Carlo (MCMC)


There are some classic methods for determining the unknown parameters in reliability analysis including probability plotting, least square, and maximum likelihood estimation (MLE).
These methods provide a simple value for parameters based on experimental data.
Bayesian approach is the method of choice employed widely in research for estimating and updating parameters values.

The advantages of this method:
1. The Bayesian approach is an updating network which does not ignore the prior information in contrast with the classic methods, but updates the earlier estimations with new obtained knowledge to improve the estimated parameter.
2. The output of Bayesian network is a distribution instead of a simple value for the parameter.
3. In classic methods, a large sample size is required for convergence of the estimate. For analysis of cases restricted by limited data the Bayesian approach is a good approach for parameter estimation.

By considering X as the unknown parameter and E as the new knowledge of crack length, Bayesian theorem modifies a prior probability ASQ-RD-Dec2015-Newsletter - Google Chrome_2 yielding a posterior probability ASQ-RD-Dec2015-Newsletter - Google Chrome_3, via the expression:
ASQ-RD-Dec2015-Newsletter - Google Chrome

where ASQ-RD-Dec2015-Newsletter - Google Chrome_4 is the likelihood function and is constructed based on new available knowledge and evidence.
The factor f(E|X)/∫f(E|X)π0(X)d(X) is the impact of the evidence on the belief in the PDF of the parameters.
Multiplying the prior PDF of the parameters by this factor provides a theoretical mechanism to update the prior knowledge of the parameters with the new evidence.

A Bayesian network is a complicated method in practice. An analytical solution rarely occurs.
However, it is possible to moderate the Bayesian difficulty by numerical method through Markov chain Monte Carlo (MCMC) solution.
This numerical method is applicable for similar approaches which need to integrate over the posterior distribution to make inference about model parameters or to make predictions.
MCMC is Monte Carlo integration that draws samples from the required distribution by running a properly constructed Markov chain for a long time.
Gibbs sampling is usually used for taking samples.
BUGs is an acronym stand for Bayesian inference using Gibbs sampling with WinBUGS as an open source software package for performing MCMC simulation.

By: Mohammad Pourgol-Mohammad, Ph.D, P.E, CRE,

Previously published in the December 2015 Volume 6, Issue 4 ASQ Reliability Division Newsletter

Picture © B. Poncelet

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