Ergodicity of an Adaptive MCMC Sampler under a Probability Bound

TL;DR

This paper establishes new, more practical conditions for ensuring the ergodicity of adaptive MCMC algorithms, allowing for unbounded target distributions without requiring artificial compactness assumptions.

Contribution

It introduces sufficient conditions based on probability bounds that relax the need for compactness in proving ergodicity of adaptive MCMC methods.

Findings

Conditions for ergodicity are applicable to unbounded target distributions.

Relaxation of compactness assumptions to probabilistic bounds.

Ensures ergodicity without artificial compactness constraints.

Abstract

This paper provides sufficient conditions over the sequence of samples and parameters of an adaptive Markov Chain Monte Carlo (MCMC) algorithm to ensure ergodicity with respect to a target distribution that can have unbounded support. These conditions aim to make more easily usable the conditions of Containment and Diminishing Adaptation from Roberts and Rosenthal [2007] formulated over the transition kernels, without needing, as was done in other works, an artificial assumption of the compactness over both sample and parameter spaces. The paper shows that the condition of compactness can be relaxed to a more realistic bound in probability over the sequence of both samples and parameters.