A central limit theorem for adaptive and interacting Markov chains

TL;DR

This paper establishes a central limit theorem for a broad class of adaptive and interacting Markov Chain Monte Carlo algorithms, enhancing understanding of their asymptotic behavior and variance estimation.

Contribution

It extends existing results by providing a CLT for unbounded functionals in a general non-Markovian framework, applicable to various adaptive and interacting MCMC methods.

Findings

Proves a CLT for additive functionals of adaptive/interacting MCMC.

Identifies the asymptotic variance for these algorithms.

Demonstrates applicability to the interacting tempering algorithm.

Abstract

Adaptive and interacting Markov Chains Monte Carlo (MCMC) algorithms are a novel class of non-Markovian algorithms aimed at improving the simulation efficiency for complicated target distributions. In this paper, we study a general (non-Markovian) simulation framework covering both the adaptive and interacting MCMC algorithms. We establish a Central Limit Theorem for additive functionals of unbounded functions under a set of verifiable conditions, and identify the asymptotic variance. Our result extends all the results reported so far. An application to the interacting tempering algorithm (a simplified version of the equi-energy sampler) is presented to support our claims.