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Ergodicity of an Adaptive MCMC Sampler under a Probability Bound

Alexandre Chotard

TL;DR

This work tackles the ergodicity of adaptive MCMC on unbounded spaces by replacing compactness assumptions with a probability-bound containment framework, leveraging a coupling-based approach. It splits the adaptation into (θ, ψ) and imposes continuity-based conditions on the kernel’s density and singular parts to guarantee containment and diminishing adaptation without requiring global compactness. The main theorem shows ergodicity relative to $π$ under these conditions, with concrete corollaries for Adaptive Metropolis-Hastings and CMA-ES–inspired rank-one MH-CMA updates. The applications demonstrate how to verify the abstract conditions via boundedness in probability and diminishing adaptation, enabling provable long-run sampling accuracy for practical adaptive schemes.

Abstract

This paper provides sufficient conditions over the sequence of samples and parameters of an adaptive Markov Chain Monte Carlo (MCMC) algorithm to converge to the target distribution. These conditions aim to make more easily usable classical conditions formulated over the transition kernels, without needing, as was done in other works, to assume the compactness of both sample and parameter spaces. The condition of compactness is replaced here with a probability bound over the sequence of both samples and parameters.

Ergodicity of an Adaptive MCMC Sampler under a Probability Bound

TL;DR

This work tackles the ergodicity of adaptive MCMC on unbounded spaces by replacing compactness assumptions with a probability-bound containment framework, leveraging a coupling-based approach. It splits the adaptation into (θ, ψ) and imposes continuity-based conditions on the kernel’s density and singular parts to guarantee containment and diminishing adaptation without requiring global compactness. The main theorem shows ergodicity relative to under these conditions, with concrete corollaries for Adaptive Metropolis-Hastings and CMA-ES–inspired rank-one MH-CMA updates. The applications demonstrate how to verify the abstract conditions via boundedness in probability and diminishing adaptation, enabling provable long-run sampling accuracy for practical adaptive schemes.

Abstract

This paper provides sufficient conditions over the sequence of samples and parameters of an adaptive Markov Chain Monte Carlo (MCMC) algorithm to converge to the target distribution. These conditions aim to make more easily usable classical conditions formulated over the transition kernels, without needing, as was done in other works, to assume the compactness of both sample and parameter spaces. The condition of compactness is replaced here with a probability bound over the sequence of both samples and parameters.
Paper Structure (13 sections, 15 theorems, 27 equations)

This paper contains 13 sections, 15 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Setting and Notations
  3. Notations
  4. Setting
  5. Main Results
  6. Conditions
  7. Continuity Results
  8. Containment
  9. Diminishing Adaptation
  10. Ergodicity of an adaptive MCMC
  11. Applications
  12. Adaptive Metropolis-Hastings
  13. Rank-one Metropolis-Hastings Covariance Matrix Adaptation

Key Result

Proposition 1

Let $(\mathcal{X}, \Sigma)$ be a $\sigma$-locally compact Hausdorff measurable space equipped with a locally finite measure $\lambda$, and let $\Gamma$ be a locally compact Hausdorff space. Let $(\mu_{\gamma})_{\gamma \in \Gamma}$ be a family of finite measures absolutely continuous with respect to

Theorems & Definitions (30)

  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 20 more