Optimal Sample Size Determination based on Bayesian Reliability and Value of Information

Abstract

In the structural reliability analysis, the probabilistic distributions of basic random variables may contain uncertainties arising from the imperfect knowledge from which the distributions are elicited. It subsequently introduces uncertainty into the calculated failure probability Pf, which may affect the decision-making. To reduce the uncertainty of the failure probability estimation, it is desirable to collect samples of the basic random variables and use these samples to update the corresponding probability distributions. In this work, the relationship between the sample size of the basic random variable and variance of the estimated failure probability is derived by using the Bayesian pre-posterior analysis, based on which the optimal sample size criterion is established. To make the pre-posterior analysis and criterion applicable to a wide range of distributions, continuous random variables are discretized at first. The probability mass functions of the discretized random variables are then assigned Dirichlet prior distributions. The total probability theorem is employed to express Pf in terms of PMFs of the discretized variables and conditional failure probabilities corresponding to given values of discretized variables. Then the prior, posterior and pre-posterior analysis of Pf are carried out. The optimal sample size criterion to maximize the expected net gain of sampling is developed based on the result of the pre-posterior analysis of Pf and quadratic loss function. An example of determining the optimal number of burst tests for collecting the samples of model error of the burst capacity model for corroded pipelines is used to illustrate the proposed criterion. Moreover, the sensitivity analysis indicates that the optimal sample size is insensitive to the discretization of the basic random variables, but sensitive to the equivalent sample size of the prior Dirichlet distribution.