Markov chain Monte Carlo (MCMC) methods provide consistent approximations of integrals as the number of iterations goes to infinity. MCMC estimators are generally biased after any fixed number of iterations, which complicates both parallel computation and the construction of confidence intervals. We propose to remove this bias by using couplings of Markov chains together with a telescopic sum argument of Glynn & Rhee (2014). The resulting unbiased estimators can be computed independently in parallel, and confidence intervals can be directly constructed from the Central Limit Theorem for i.i.d. variables. We discuss practical couplings for popular algorithms such as the Metropolis-Hastings, Gibbs, and Hamiltonian Monte Carlo samplers. We establish the theoretical validity of the proposed estimators and study their efficiency compared to the underlying MCMC algorithms. The method is illustrated on toy examples, on a variable selection problem and on the approximation of the cut distribution.
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