Forgetting the starting distribution in finite interacting tempering

TL;DR

This paper proves that the interacting tempering adaptive MCMC algorithm quickly forgets its initial state under certain conditions, but this does not always guarantee convergence to the target distribution, raising concerns about convergence diagnostics.

Contribution

It provides a coupling-based proof of rapid forgetting in interacting tempering and highlights limitations of using this property as a convergence diagnostic.

Findings

Rapid forgetting under certain assumptions

Applicable to models like Ising, Potts, and random graphs

Counterexample shows slow convergence despite forgetting

Abstract

Markov chain Monte Carlo (MCMC) methods are frequently used to approximately simulate high-dimensional, multimodal probability distributions. In adaptive MCMC methods, the transition kernel is changed "on the fly" in the hope to speed up convergence. We study interacting tempering, an adaptive MCMC algorithm based on interacting Markov chains, that can be seen as a simplified version of the equi-energy sampler. Using a coupling argument, we show that under easy to verify assumptions on the target distribution (on a finite space), the interacting tempering process rapidly forgets its starting distribution. The result applies, among others, to exponential random graph models, the Ising and Potts models (in mean field or on a bounded degree graph), as well as (Edwards-Anderson) Ising spin glasses. As a cautionary note, we also exhibit an example of a target distribution for which the interacting tempering process rapidly forgets its starting distribution, but takes an exponential number of steps (in the dimension of the state space) to converge to its limiting distribution. As a consequence, we argue that convergence diagnostics that are based on demonstrating that the process has forgotten its starting distribution might be of limited use for adaptive MCMC algorithms like interacting tempering.