An MCMC Method to Sample from Lattice Distributions

We introduce a Markov Chain Monte Carlo (MCMC) algorithm to generate samples from probability distributions supported on a d-dimensional lattice Lambda = BmathbbZd, where B is a full-rank matrix. Specifically, we consider lattice distributions P_Lambda in which the probability at a lattice point is proportional to a given probability density function, f, evaluated at that point. To generate samples from P_Lambda, it suffices to draw samples from a pull-back measure P_mathbbZd defined on the integer lattice. The probability of an integer lattice point under P_mathbbZd is proportional to the density function π = |det(B)|fcirc B. The algorithm we present in this paper for sampling from P_mathbbZd is based on the Metropolis-Hastings framework. In particular, we use π as the proposal distribution and calculate the Metropolis-Hastings acceptance ratio for a well-chosen target distribution. We can use any method, denoted by ALG, that ideally draws samples from the probability density π, to generate a proposed state. The target distribution is a piecewise sigmoidal distribution, chosen such that the coordinate-wise rounding of a sample drawn from the target distribution gives a sample from P_mathbbZd. When ALG is ideal, we show that our algorithm is uniformly ergodic if -log(π) satisfies a gradient Lipschitz condition.