Unbiased Hamiltonian Monte Carlo with couplings
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We propose a coupling approach to parallelize Hamiltonian Monte Carlo estimators, following Jacob, O'Leary & Atchadé (2017). A simple coupling, obtained by using common initial velocities and common uniform variables for the acceptance steps, leads to pairs of Markov chains that contract, in the sense that the distance between them can become arbitrarily small. We show how this strategy can be combined with coupled random walk Metropolis-Hastings steps to enable exact meetings of the two chains, and in turn, unbiased estimators that can be computed in parallel and averaged. The resulting estimator is valid in the limit of the number of independent replicates, instead of the usual limit of the number of Markov chain iterations. We investigate the effect of tuning parameters, such as the number of leap-frog steps and the step size, on the estimator's efficiency. The proposed methodology is demonstrated on a 250-dimensional Normal distribution, on a bivariate Normal truncated by linear and quadratic inequalities, and on a logistic regression with 300 covariates.
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