  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.
