Probability measure changes in Monte Carlo simulation

Abstract

The objective of Bayesian inference is often to infer, from data, a probability measure for a random variable that can be used as input for Monte Carlo simulation. When datasets for Bayesian inference are small, a principle challenge is that, as additional data are collected, the probability measure inferred from Bayesian inference may change significantly. That is, the original probability density inferred from Bayesian inference may differ considerably from the updated probability density both in its model form and parameters. In such cases, expensive Monte Carlo simulations may have already been performed using the original distribution and it is infeasible to start again and perform a new Monte Carlo analysis using the updated density due to the large added computational cost. In this work, we discuss four strategies for updating Mote Carlo simulations for such a change in probability measure: 1. Importance sampling reweighting; 2. A sample augmenting strategy; 3. A sample filtering strategy; and 4. A mixed augmenting-filtering strategy. The efficiency of each strategy is compared and the ultimate aim is to achieve the change in distribution with a minimal number of added computational simulations. The comparison results show that when the change in measure is small importance sampling reweighting can be very effective. Otherwise, a proposed novel mixed augmenting-filtering algorithm can robustly and efficiently accommodate a measure change in Monte Carlo simulation that minimizes the impact on the sample set and saves a large amount of additional computational cost. The strategy is then applied for uncertainty quantification in the buckling strength of a simple plate given ongoing data collection to estimate uncertainty in the yield stress.