Principle of detailed balance and convergence assessment of Markov Chain Monte Carlo methods and simulated annealing

TL;DR

This paper introduces a convergence diagnostic for Markov Chain Monte Carlo methods based on detailed balance, providing an intuitive and practical tool for assessing convergence and comparing algorithm efficiency.

Contribution

It proposes a new qualitative convergence criterion rooted in detailed balance, applicable under weak conditions, and demonstrates its use in simulated annealing and sampling tasks.

Findings

The criterion effectively assesses convergence in various sampling scenarios.

It can serve as a stopping rule for simulated annealing.

A new measure of algorithm efficiency is derived from the criterion.

Abstract

Markov Chain Monte Carlo (MCMC) methods are employed to sample from a given distribution of interest, whenever either the distribution does not exist in closed form, or, if it does, no efficient method to simulate an independent sample from it is available. Although a wealth of diagnostic tools for convergence assessment of MCMC methods have been proposed in the last two decades, the search for a dependable and easy to implement tool is ongoing. We present in this article a criterion based on the principle of detailed balance which provides a qualitative assessment of the convergence of a given chain. The criterion is based on the behaviour of a one-dimensional statistic, whose asymptotic distribution under the assumption of stationarity is derived; our results apply under weak conditions and have the advantage of being completely intuitive. We implement this criterion as a stopping rule for simulated annealing in the problem of finding maximum likelihood estimators for parameters of a 20-component mixture model. We also apply it to the problem of sampling from a 10-dimensional funnel distribution via slice sampling and the Metropolis-Hastings algorithm. Furthermore, based on this convergence criterion we define a measure of efficiency of one algorithm versus another.