Kinetic Theories for Metropolis Monte Carlo Methods

TL;DR

This paper develops kinetic theories for the Metropolis Monte Carlo algorithm across various scaling regimes, offering new insights and modifications to improve its application in Bayesian inverse problems.

Contribution

It introduces kinetic equations for Metropolis Monte Carlo in different regimes, providing a novel perspective and practical modifications for Bayesian inverse problems.

Findings

Derived kinetic equations for Metropolis Monte Carlo.

Proposed modifications based on different scaling regimes.

Validated improvements through simulation of the Lorenz system.

Abstract

We consider generalizations of the classical inverse problem to Bayesien type estimators, where the result is not one optimal parameter but an optimal probability distribution in parameter space. The practical computational tool to compute these distributions is the Metropolis Monte Carlo algorithm. We derive kinetic theories for the Metropolis Monte Carlo method in different scaling regimes. The derived equations yield a different point of view on the classical algorithm. It further inspired modifications to exploit the difference scalings shown on an simulation example of the Lorenz system.