Variable transformation to obtain geometric ergodicity in the random-walk Metropolis algorithm

TL;DR

This paper introduces a variable transformation method that ensures geometric ergodicity of the random-walk Metropolis algorithm for a broader class of target distributions, including many Bayesian posterior distributions.

Contribution

The authors propose a change-of-variable approach to transform target densities, enabling geometric ergodicity in cases where it previously did not hold, expanding the algorithm's applicability.

Findings

Method successfully achieves geometric ergodicity for non-super-exponentially light distributions.

Applicable to Bayesian models with logistic, Poisson, and log-linear posteriors.

Widely applicable transformation technique enhances MCMC convergence properties.

Abstract

A random-walk Metropolis sampler is geometrically ergodic if its equilibrium density is super-exponentially light and satisfies a curvature condition [Stochastic Process. Appl. 85 (2000) 341-361]. Many applications, including Bayesian analysis with conjugate priors of logistic and Poisson regression and of log-linear models for categorical data result in posterior distributions that are not super-exponentially light. We show how to apply the change-of-variable formula for diffeomorphisms to obtain new densities that do satisfy the conditions for geometric ergodicity. Sampling the new variable and mapping the results back to the old gives a geometrically ergodic sampler for the original variable. This method of obtaining geometric ergodicity has very wide applicability.