An MCMC Method to Sample from Lattice Distributions

• URL: http://arxiv.org/abs/2101.06453v2
• Date: Tue, 26 Jan 2021 12:23:37 GMT
• Title: An MCMC Method to Sample from Lattice Distributions
• Authors: Anand Jerry George, Navin Kashyap
• Abstract summary: We introduce a Markov Chain Monte Carlo algorithm to generate samples from probability distributions supported on a $d$-dimensional lattice $Lambda. We use $pi$ as the proposal distribution and calculate the Metropolis-Hastings acceptance ratio for a well-chosen target distribution.
• Score: 4.4044968357361745
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce a Markov Chain Monte Carlo (MCMC) algorithm to generate samples from probability distributions supported on a $d$-dimensional lattice $\Lambda = \mathbf{B}\mathbb{Z}^d$, where $\mathbf{B}$ 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 $P_{\mathbb{Z}^d}$ defined on the integer lattice. The probability of an integer lattice point under $P_{\mathbb{Z}^d}$ is proportional to the density function $\pi = |\det(\mathbf{B})|f\circ \mathbf{B}$. 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 $\pi$ 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 $\pi$, 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 $-\log(\pi)$ satisfies a gradient Lipschitz condition.

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