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Abstract
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MCMC trajectories resemble diffusion paths.
Such an interpretation has been substantiated in earlier works in the literature when it has been proven that, when the dimensionality of the state space increases, the MCMC trajectory converges to a particular diffusion process.
Such a result, though proven in simplified scenaria of iid targets, provides insight in the behavior of MCMC algorithms in high dimensions.
We examine the case of the so-called "hybrid Monte-Carlo" MCMC algorithm, invoking Hamiltonian dynamics, employed by physicists in molecular dynamics applications and elsewhere.
Bridging the machinery employed above with tools from numerical analysis we show that the MCMC trajectory of the hybrid algorithm converges (when appropriately rescaled) to a hypoelliptic SDE. Such a result provides a complete characterization of the efficiency of the algorithm: we conclude that the hybrid algorithm should be scaled as 1/n^{1/6} (n being the dimensionality) , with optimal acceptance probability 0.743.
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