Unbiased Markov chain Monte Carlo with couplings

Abstract

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.

arXiv commentFinal version, accepted as a JRSS discussion paper; includes supplementary material as appendices; 12 figures, 48 pages

References in corpus (23)

• Rejoinder to "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Least Angle Regression
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• MCMC using Hamiltonian dynamics
• General state space Markov chains and MCMC algorithms
• Elliptical slice sampling
• On risk bounds in isotonic and other shape restricted regression problems
• Geometric integrators and the Hamiltonian Monte Carlo method
• Rapid Mixing of Hamiltonian Monte Carlo on Strongly Log-Concave Distributions
• Piecewise-Deterministic Markov Chain Monte Carlo
• Parallelizing MCMC with Random Partition Trees
• Better together? Statistical learning in models made of modules
• Exact sampling for intractable probability distributions via a Bernoulli factory
• Unbiased estimation of log normalizing constants with applications to Bayesian cross-validation
• Circularly-Coupled Markov Chain Sampling
• Coupling and Decoupling to bound an approximating Markov Chain

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• Semi-Modular Inference: enhanced learning in multi-modular models by tempering the influence of components
• Regeneration-enriched Markov processes with application to Monte Carlo
• Accelerated Sampling on Discrete Spaces with Non-Reversible Markov Processes
• Unbiased estimation of log normalizing constants with applications to Bayesian cross-validation
• Estimating Convergence of Markov chains with L-Lag Couplings
• SUMO: Unbiased Estimation of Log Marginal Probability for Latent Variable Models
• Distributed Computation for Marginal Likelihood based Model Choice
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