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arXiv · 2508.05303

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

Emil Løvbak·Sebastian Krumscheid
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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.

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Emil Løvbak, Sebastian Krumscheid. 2025-08-07. An Investigation into the Distribution of Ratios of Particle Solver-based Likelihoods. https://arxiv.org/abs/2508.05303

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