# Comparison of Markov chains via weak Poincar'e inequalities with application to pseudo-marginal MCMC

arXiv: https://arxiv.org/abs/2112.05605 · HTML: https://mlsanity.com/paper/2112.05605/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 41 references that are in the map.
- First posted on arXiv 10 Dec 2021.

## Builds on (most cited first, 1 of 1)

- [The Block Pseudo-Marginal Sampler](https://mlsanity.com/paper/1603.02485/) (2016)

## Built on by (most cited first, 5 of 5)

- [A Separation in Heavy-Tailed Sampling: Gaussian vs. Stable Oracles for Proximal Samplers](https://mlsanity.com/paper/2405.16736/) (2024)
- [Mean-Square Analysis of Discretized Ito Diffusions for Heavy-tailed Sampling](https://mlsanity.com/paper/2303.00570/) (2023)
- [Fast convergence of the Expectation Maximization algorithm under a logarithmic Sobolev inequality](https://mlsanity.com/paper/2407.17949/) (2024)
- [Non-asymptotic bounds for forward processes in denoising diffusions: Ornstein-Uhlenbeck is hard to beat](https://mlsanity.com/paper/2408.13799/) (2024)
- [Local and Global Contraction Principles for MCMC Mixing](https://mlsanity.com/paper/2606.03033/) (2026)

## Abstract (arXiv)

We investigate the use of a certain class of functional inequalities known as weak Poincar'e 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).