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Comparison of Markov chains via weak Poincaré inequalities with application to pseudo-marginal MCMC

Christophe Andrieu, Anthony Lee, Sam Power, Andi Q. Wang

TL;DR

This work investigates the use of a certain class of functional inequalities known as weak Poincar\'e inequalities to bound convergence of Markov chains to equilibrium and shows that this enables the straightforward and transparent derivation of subgeometric convergence bounds for methods such as the Independent Metropolis--Hastings sampler and pseudo-marginal methods for intractable likelihoods.

Abstract

We investigate the use of a certain class of functional inequalities known as weak Poincaré inequalities to bound convergence of Markov chains to equilibrium. We show that this enables the straightforward and transparent derivation of subgeometric convergence bounds for methods such as the Independent Metropolis--Hastings sampler and pseudo-marginal methods for intractable likelihoods, the latter being subgeometric in many practical settings. These results rely on novel quantitative comparison theorems between Markov chains. Associated proofs are simpler than those relying on drift/minorization conditions and the tools developed allow us to recover and further extend known results as particular cases. We are then able to provide new insights into the practical use of pseudo-marginal algorithms, analyse the effect of averaging in Approximate Bayesian Computation (ABC) and the use of products of independent averages, and also to study the case of lognormal weights relevant to particle marginal Metropolis--Hastings (PMMH).

Comparison of Markov chains via weak Poincaré inequalities with application to pseudo-marginal MCMC

TL;DR

This work investigates the use of a certain class of functional inequalities known as weak Poincar\'e inequalities to bound convergence of Markov chains to equilibrium and shows that this enables the straightforward and transparent derivation of subgeometric convergence bounds for methods such as the Independent Metropolis--Hastings sampler and pseudo-marginal methods for intractable likelihoods.

Abstract

We investigate the use of a certain class of functional inequalities known as weak Poincaré inequalities to bound convergence of Markov chains to equilibrium. We show that this enables the straightforward and transparent derivation of subgeometric convergence bounds for methods such as the Independent Metropolis--Hastings sampler and pseudo-marginal methods for intractable likelihoods, the latter being subgeometric in many practical settings. These results rely on novel quantitative comparison theorems between Markov chains. Associated proofs are simpler than those relying on drift/minorization conditions and the tools developed allow us to recover and further extend known results as particular cases. We are then able to provide new insights into the practical use of pseudo-marginal algorithms, analyse the effect of averaging in Approximate Bayesian Computation (ABC) and the use of products of independent averages, and also to study the case of lognormal weights relevant to particle marginal Metropolis--Hastings (PMMH).
Paper Structure (28 sections, 39 theorems, 209 equations, 1 figure)

This paper contains 28 sections, 39 theorems, 209 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Notation
  4. Weak Poincaré inequalities
  5. General case
  6. Definitions and basic results
  7. Examples of $\beta\left(s\right)$ and $\gamma=F_{a}^{-1}$
  8. Reversible case
  9. Simplified weak Poincaré inequality
  10. Necessity of weak Poincaré inequalities
  11. Illustration: Independent MH sampler
  12. Exponential target and proposal case
  13. Polynomial target and proposal case
  14. Chaining Poincaré inequalities
  15. Application to pseudo-marginal methods
  16. ...and 13 more sections

Key Result

Lemma 7

Let $F_{a}(\cdot)\colon(0,a]\rightarrow\mathbb{R}$, where $(0,a]\subset\mathrm{D}$, be given by where $K^{*}$ is given in Definition def:KandKstar and $\mathrm{D}:=\{v\ge0:K^{*}(v)<\infty\}$. Then $F_{a}(\cdot)$

Figures (1)

  • Figure 1: A plot of the function $\sigma\mapsto\log\left(\tilde{v}(\sigma)/\sigma^{2}\right)$ in the case $C_{\mathrm{P}}=1$.

Theorems & Definitions (98)

  • Definition 1
  • Remark 2
  • Remark 3
  • Definition 4
  • Remark 5
  • Definition 6
  • Lemma 7
  • Theorem 8
  • Remark 9
  • Remark 10
  • ...and 88 more