An Investigation into the Distribution of Ratios of Particle Solver-based Likelihoods

TL;DR

This paper examines how particle-based Monte Carlo methods influence the distribution of likelihood ratios in Bayesian inverse problems, impacting the acceptance step in Metropolis-Hastings sampling, through theoretical and numerical analysis.

Contribution

It provides a novel analysis of the distribution of likelihood ratios when using particle-based likelihood approximations in Bayesian sampling.

Findings

Likelihood ratios are affected by Gaussian noise in the approximate likelihoods.

Theoretical insights into the distribution of likelihood ratios under particle-based approximations.

Numerical experiments validate the theoretical analysis.

Abstract

We investigate the use of the Metropolis-Hastings algorithm to sample posterior distribution in a Bayesian inverse problem, where the likelihood function is random. Concretely, we consider the case where one has full field observations of a PDE solution, in case a one-dimensional diffusion equation, subject to a Gaussian observation error. Assuming one uses a particle-based Monte Carlo simulation when approximating the likelihood function, one gets an approximate likelihood with additive Gaussian noise in the log-likelihood. We study how these two Gaussian distributions affect the distribution of ratios of approximate likelihood evaluations, as required when evaluating acceptance probabilities in the Metropolis-Hastings algorithm. We do so through both theoretical analysis and numerical experiments.