Markov Chain Monte Carlo (MCMC) algorithms play an important role instatistical inference problems dealing with intractable probabilitydistributions. Recently, many MCMC algorithms such as Hamiltonian Monte Carlo(HMC) and Riemannian Manifold HMC have been proposed to provide distantproposals with high acceptance rate. These algorithms, however, tend to becomputationally intensive which could limit their usefulness, especially forbig data problems due to repetitive evaluations of functions and statisticalquantities that depend on the data. This issue occurs in many statisticcomputing problems. In this paper, we propose a novel strategy that exploitssmoothness (regularity) of parameter space to improve computational efficiencyof MCMC algorithms. When evaluation of functions or statistical quantities areneeded at a point in parameter space, interpolation from precomputed values orprevious computed values is used. More specifically, we focus on HamiltonianMonte Carlo (HMC) algorithms that use geometric information for fasterexploration of probability distributions. Our proposed method is based onprecomputing the required geometric information on a set of grids beforerunning sampling information at nearby grids at each iteration of HMC. Sparsegrid interpolation method is used for high dimensional problems. Tests oncomputational examples are shown to illustrate the advantages of our method.
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