MCMC and its applications to bayesian updating and sensitivity analysis




Vazquez, Eric Giovanni

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Markov Chain Monte Carlo (MCMC) sampling is a powerful and popular tool for generating samples needed in computing high-dimensional integrals, particularly for Bayesian inferencing. This thesis looks at the application of MCMC with the Score Function sensitivity method to compute probabilistic sensitivities. The thesis also looks at the MCMC applied to Bayesian updating to aid in the life prediction and structural reliability of rotorcraft or any fatigue prone structure. The results using the Score Function sensitivity method indicate that MCMC converges to the correct sensitivity but at a slower rate than standard Monte Carlo (MC) sampling. That is, the variance of the sensitivities is larger for MCMC than MC. As a result, analytical variance estimates for the sensitivities applicable to MC are not accurate for MCMC. Hence, the Score Function method must be applied with care when combined with MCMC.

Bayesian updating involves combining prior probability estimates with observed data to create a model for the distribution of the corresponding field data. For example, using a prior probability estimate for the initial flaw size of a fatigue prone material is practical because it can be applied quickly based on experience with the structure, but these estimates are rarely precise. Building a distribution from observed data is a more accurate means of obtaining this information because it is derived from actual data; however, gathering enough data to create a reliable distribution would be too cumbersome and expensive. Combining prior probability estimates with observed data makes Bayesian updating a time and cost efficient tool for building an accurate flaw size probability density function. For many practical applications of Bayesian updating, the computations are still very time and resource intensive; thus, there exists a need for a more efficient way to conduct the computations while still maintaining a high degree of accuracy. By applying MCMC to Bayesian updating, it is possible to eliminate the computations that involve integrating complicated, high dimensional functions that require a large amount of time and computer memory. A few numerical examples are used to demonstrate the updating of the parameters of the initial crack size distribution.


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Bayesian updating, Markov Chain Monte Carlo, Score Function method, Sensitivity Analysis



Mechanical Engineering