Partition function approach to non-Gaussian likelihoods: physically motivated convergence criteria for Markov chains

Abstract

ABSTRACT Non-Gaussian distributions in cosmology are commonly evaluated with Monte Carlo Markov chain methods, as the Fisher matrix formalism is restricted to the Gaussian case. The Metropolis–Hastings algorithm will provide samples from the posterior distribution after a burn-in period, and the corresponding convergence is usually quantified with the Gelman–Rubin criterion. In this paper, we investigate the convergence of the Metropolis–Hastings algorithm by drawing analogies to statistical Hamiltonian systems in thermal equilibrium for which a canonical partition sum exists. Specifically, we quantify virialization, equipartition, and thermalization of Hamiltonian Monte Carlo Markov chains for a toy model and for the likelihood evaluation for a simple dark energy model constructed from supernova data. We follow the convergence of these criteria to the values expected in thermal equilibrium, in comparison to the Gelman–Rubin criterion. We find that there is a much larger class of physically motivated convergence criteria with clearly defined target values indicating convergence. As a numerical tool, we employ physics-informed neural networks for speeding up the sampling process.