Hastings-Metropolis algorithm on Markov chains for small-probability estimation

TL;DR

This paper adapts the Hastings-Metropolis algorithm for small-probability estimation in neutron transport shielding studies involving Markov chains, demonstrating improved efficiency over traditional Monte Carlo methods.

Contribution

It introduces a novel adaptation of the Hastings-Metropolis algorithm for Markov chain-based small-probability estimation and provides convergence analysis and practical implementation details.

Findings

The adapted algorithm converges reliably for Markov chain problems.

The method significantly outperforms simple Monte Carlo in small-probability estimation.

Practical implementation shows improved efficiency in neutron shielding simulations.

Abstract

Shielding studies in neutron transport, with Monte Carlo codes, yield challenging problems of small-probability estimation. The particularity of these studies is that the small probability to estimate is formulated in terms of the distribution of a Markov chain, instead of that of a random vector in more classical cases. Thus, it is not straightforward to adapt classical statistical methods, for estimating small probabilities involving random vectors, to these neutron-transport problems. A recent interacting-particle method for small-probability estimation, relying on the Hastings-Metropolis algorithm, is presented. It is shown how to adapt the Hastings-Metropolis algorithm when dealing with Markov chains. A convergence result is also shown. Then, the practical implementation of the resulting method for small-probability estimation is treated in details, for a Monte Carlo shielding study. Finally, it is shown, for this study, that the proposed interacting-particle method considerably outperforms a simple-Monte Carlo method, when the probability to estimate is small.