# Unbiased Markov chain Monte Carlo with couplings

arXiv: https://arxiv.org/abs/1708.03625 · HTML: https://mlsanity.com/paper/1708.03625/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Aug 2017.

## Built on by (most cited first, 10 of 10)

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- [Unbiased Estimation using Underdamped Langevin Dynamics](https://mlsanity.com/paper/2206.07202/) (2022)
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- [Mixing times of data-augmentation Gibbs samplers for high-dimensional probit regression](https://mlsanity.com/paper/2505.14343/) (2025)
- [Monte Carlo Filtering Objectives: A New Family of Variational Objectives to Learn Generative Model and Neural Adaptive Proposal for Time Series](https://mlsanity.com/paper/2105.09801/) (2021)
- [Antithetic Riemannian Manifold And Quantum-Inspired Hamiltonian Monte Carlo](https://mlsanity.com/paper/2107.02070/) (2021)
- [Leveraging Nested MLMC for Sequential Neural Posterior Estimation with Intractable Likelihoods](https://mlsanity.com/paper/2401.16776/) (2024)
- [Feynman-Kac Operator Expectation Estimator](https://mlsanity.com/paper/2407.02010/) (2024)

## Abstract (arXiv)

Markov chain Monte Carlo (MCMC) methods provide consistent of integrals as the number of iterations goes to infinity. MCMC estimators are generally biased after any fixed number of iterations. We propose to remove this bias by using couplings of Markov chains together with a telescopic sum argument of Glynn and Rhee (2014). The resulting unbiased estimators can be computed independently in parallel. We discuss practical couplings for popular MCMC algorithms. We establish the theoretical validity of the proposed estimators and study their efficiency relative to the underlying MCMC algorithms. Finally, we illustrate the performance and limitations of the method on toy examples, on an Ising model around its critical temperature, on a high-dimensional variable selection problem, and on an approximation of the cut distribution arising in Bayesian inference for models made of multiple modules.