# 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 Î = đâ¤^d, where đ is a full-rank matrix. Specifically, we consider lattice distributions P_Î 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_Î, it suffices to draw samples from a pull-back measure P_â¤^d defined on the integer lattice. The probability of an integer lattice point under P_â¤^d is proportional to the density function Ď = |(đ)|fâđ. The algorithm we present in this paper for sampling from P_â¤^d is based on the Metropolis-Hastings framework. In particular, we use Ď 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 Ď, 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_â¤^d. When ALG is ideal, we show that our algorithm is uniformly ergodic if -log(Ď) satisfies a gradient Lipschitz condition.

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