# Maximal couplings of the Metropolis-Hastings algorithm

arXiv: https://arxiv.org/abs/2010.08573 · HTML: https://mlsanity.com/paper/2010.08573/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Oct 2020.

## Built on by (most cited first, 7 of 7)

- [Differentiating Metropolis-Hastings to Optimize Intractable Densities](https://mlsanity.com/paper/2306.07961/) (2023)
- [Bounding Wasserstein distance with couplings](https://mlsanity.com/paper/2112.03152/) (2021)
- [Optimal randomized multilevel Monte Carlo for repeatedly nested expectations](https://mlsanity.com/paper/2301.04095/) (2023)
- [Unbiased Estimation using Underdamped Langevin Dynamics](https://mlsanity.com/paper/2206.07202/) (2022)
- [Linear-cost unbiased posterior estimates for crossed effects and matrix factorization models via couplings](https://mlsanity.com/paper/2410.08939/) (2024)
- [End-to-end Training of Deep Boltzmann Machines by Unbiased Contrastive Divergence with Local Mode Initialization](https://mlsanity.com/paper/2305.19684/) (2023)
- [GradInf: Gradient Estimation as Probabilistic Inference](https://mlsanity.com/paper/2607.07840/) (2026)

## Abstract (arXiv)

Couplings play a central role in the analysis of Markov chain Monte Carlo algorithms and appear increasingly often in the algorithms themselves, e.g. in convergence diagnostics, parallelization, and variance reduction techniques. Existing couplings of the Metropolis-Hastings algorithm handle the proposal and acceptance steps separately and fall short of the upper bound on one-step meeting probabilities given by the coupling inequality. This paper introduces maximal couplings which achieve this bound while retaining the practical advantages of current methods. We consider the properties of these couplings and examine their behavior on a selection of numerical examples.