Bayesian Updating and Uncertainty Quantification using Sequential Tempered MCMC with the Rank-One Modified Metropolis Algorithm

Abstract

Bayesian methods are critical for quantifying the behaviors of systems. They capture our uncertainty about a system's behavior using probability distributions and update this understanding as new information becomes available. Probabilistic predictions that incorporate this uncertainty can then be made to evaluate system performance and make decisions. While Bayesian methods are very useful, they are often computationally intensive. This necessitates the development of more efficient algorithms. Here, we discuss a group of population Markov Chain Monte Carlo (MCMC) methods for Bayesian updating and system reliability assessment that we call Sequential Tempered MCMC (ST-MCMC) algorithms. These algorithms combine 1) a notion of tempering to gradually transform a population of samples from the prior to the posterior through a series of intermediate distributions, 2) importance resampling, and 3) MCMC. They are a form of Sequential Monte Carlo and include algorithms like Transitional Markov Chain Monte Carlo and Subset Simulation. We also introduce a new sampling algorithm called the Rank-One Modified Metropolis Algorithm (ROMMA), which builds upon the Modified Metropolis Algorithm used within Subset Simulation to improve performance in high dimensions. Finally, we formulate a single algorithm to solve combined Bayesian updating and reliability assessment problems to make posterior assessments of system reliability. The algorithms are then illustrated by performing prior and posterior reliability assessment of a water distribution system with unknown leaks and demands.