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 $Λ= \mathbf{B}\mathbb{Z}^d$, where $\mathbf{B}$ is a full-rank matrix. Specifically, we consider lattice distributions $P_Λ$ 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 $P_Λ$, it suffices to draw samples from a pull-back measure $P_{\mathbb{Z}^d}$ defined on the integer lattice. The probability of an integer lattice point under $P_{\mathbb{Z}^d}$ is proportional to the density function $π= |\det(\mathbf{B})|f\circ \mathbf{B}$. The algorithm we present in this paper for sampling from $P_{\mathbb{Z}^d}$ 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 $P_{\mathbb{Z}^d}$. When ALG is ideal, we show that our algorithm is uniformly ergodic if $-\log(π)$ satisfies a gradient Lipschitz condition.