Interacting Markov chain Monte Carlo methods for solving nonlinear measure-valued equations

TL;DR

This paper introduces a new class of interacting Markov chain Monte Carlo algorithms that leverage self-interacting processes to efficiently sample from complex, evolving probability measures, with proven convergence properties.

Contribution

The paper develops a novel theoretical framework for self-interacting MCMC algorithms that adapt based on past occupation measures, enabling sampling from sequences of complex target distributions.

Findings

Proven exponential convergence estimates.

Established uniform convergence across multiple target measures.

Illustrated effectiveness in Feynman-Kac distribution flows.

Abstract

We present a new class of interacting Markov chain Monte Carlo algorithms for solving numerically discrete-time measure-valued equations. The associated stochastic processes belong to the class of self-interacting Markov chains. In contrast to traditional Markov chains, their time evolutions depend on the occupation measure of their past values. This general methodology allows us to provide a natural way to sample from a sequence of target probability measures of increasing complexity. We develop an original theoretical analysis to analyze the behavior of these iterative algorithms which relies on measure-valued processes and semigroup techniques. We establish a variety of convergence results including exponential estimates and a uniform convergence theorem with respect to the number of target distributions. We also illustrate these algorithms in the context of Feynman-Kac distribution flows.