Bayesian Post-Processing for Subset Simulation for Decision Making Under Risk

Abstract eng:
One the most important and computationally challenging problems in reliability engineering is to estimate the failure probability, that is, the probability of unacceptable system performance. In cases of practical interest, the failure probability is given by an integral over a high-dimensional uncertain parameter space. Over the past decade, the engineering research community has realized the importance of advanced stochastic simulation methods for solving reliability problems. Subset Simulation (SS), proposed by Au and Beck 1, provides an efficient algorithm for computing failure probabilities for general high-dimensional reliability problems. In this work a Bayesian post-processor for the original Subset Simulation method is developed. Monte Carlo methods are consistent with a purely frequentist procedure, meaning that they can be interpreted in terms of the frequentist definition of probability which identifies it with the long-run relative frequency of occurrence of an event. An alternative is the Bayesian approach which views probability as a measure of the plausibility of a proposition when incomplete information does not allow us to establish its truth or falsehood with certainty. In the Bayesian post-processor for Subset Simulation (SS+), the uncertain failure probability that one is estimating is modeled as a stochastic variable whose possible values belong to the unit interval. Simulated samples are viewed as informative data relevant to the system's reliability. Instead of a single real number as an estimate, SS+ produces the posterior PDF of the failure probability, which takes into account both prior information and the sampled data. This PDF can be used to give a point estimate of the failure probability such as the MAP (maximum a posterior) value, or, alternatively, can be fully used (e.g. for the life-cycle cost analysis, decision making, etc.). The relationship between the original SS estimate and SS+ is also discussed. It is shown, for example, that the failure probability estimate from the original SS can be viewed as a maximum likelihood estimate based on the sampled data only and so ignores any prior information about the failure probability.

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