Note on the convergence time of some non-reversible Markov chain Monte Carlo methods
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Introduced in statistical physics, non-reversible Markov chain Monte Carlo algorithms (MCMC) have recently received an increasing attention from the computational statistics community. The main motivation is that, in the context of MCMC algorithms, non-reversible Markov chains usually yield more accurate empirical estimators than their reversible counterparts, such as those obtained using the Metropolis-Hastings algorithm. In this note, we study the efficiency of non-reversible MCMC algorithms according to their mixing time, i.e. the time the underlying Markov chain typically takes to converge to its stationary distribution. In particular, we explore the potential conflict between mixing time and asymptotic efficiency of some non-reversible MCMC algorithms. This point, which is overlooked by the existing literature, has obvious practical implications. We accompany our analysis with an novel non-reversible MCMC algorithm that aims at solving, in some capacity, this conflict. This is achieved by simultaneously introducing a skew-symmetric perturbation in the Metropolis-Hastings ratio and enlarging the state-space with an auxiliary momentum variable. We give illustrations of its efficiency on several discrete examples.
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