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

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.

💡 Research Summary

This paper investigates the impact of stochastic errors arising from particle‑based Monte Carlo simulations on Bayesian inverse problems, focusing on the Metropolis‑Hastings (MH) algorithm. The authors consider a one‑dimensional diffusion equation with periodic boundary conditions and full‑field observations corrupted by Gaussian noise. The forward model, which maps the diffusion coefficient D to the solution field, is approximated by a Monte Carlo particle ensemble. The discretization introduces two sources of error: a spatial‑binning bias that decays as O(Δx²) and a sampling variance that scales as O(1/(P·Δx)), where P is the number of particles. These errors are modeled as an additive Gaussian perturbation δ∼N(μ(D),Σ(D)) to the exact forward map G