On nonlinear Markov chain Monte Carlo

TL;DR

This paper introduces nonlinear Markov chain Monte Carlo methods that utilize nonlinear kernels to improve sampling efficiency, with theoretical guarantees and simulation results demonstrating their effectiveness.

Contribution

The paper develops a class of nonlinear MCMC algorithms using nonlinear kernels and proves their strong law of large numbers under certain conditions.

Findings

Nonlinear kernels can improve MCMC performance.

Approximations of nonlinear kernels satisfy strong law of large numbers.

Simulation results support theoretical findings.

Abstract

Let $\mathscr{P}(E)$ be the space of probability measures on a measurable space $(E,\mathcal{E})$. In this paper we introduce a class of nonlinear Markov chain Monte Carlo (MCMC) methods for simulating from a probability measure $\pi\in\mathscr{P}(E)$. Nonlinear Markov kernels (see [Feynman--Kac Formulae: Genealogical and Interacting Particle Systems with Applications (2004) Springer]) $K:\mathscr{P}(E)\times E\rightarrow\mathscr{P}(E)$ can be constructed to, in some sense, improve over MCMC methods. However, such nonlinear kernels cannot be simulated exactly, so approximations of the nonlinear kernels are constructed using auxiliary or potentially self-interacting chains. Several nonlinear kernels are presented and it is demonstrated that, under some conditions, the associated approximations exhibit a strong law of large numbers; our proof technique is via the Poisson equation and Foster--Lyapunov conditions. We investigate the performance of our approximations with some simulations.