# An MCMC Method to Sample from Lattice Distributions

George, AJ and Kashyap, N (2021) An MCMC Method to Sample from Lattice Distributions. In: 2021 IEEE International Symposium on Information Theory, ISIT 2021, 12-20 Jul 2021, Melbourne, pp. 3074-3079.

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Official URL: https://doi.org/10.1109/ISIT45174.2021.9517919

## Abstract

We introduce a Markov Chain Monte Carlo (MCMC) algorithm to generate samples from probability distributions supported on a d-dimensional lattice Λ=\textBZd, where B is a full-rank matrix. Specifically, we consider lattice distributions PΛ 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 PΛ, it suffices to draw samples from a pullback measure P\mathbbZ\mathbbd defined on the integer lattice. The probability of an integer lattice point under P\mathrmZ\mathrmd is proportional to the density function π=\vert \det(\mathrmB)\vert f\mathrmo 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\mathbbZ\mathrmd. When ALG is ideal, we show that our algorithm is uniformly ergodic if -łog(π) satisfies a gradient Lipschitz condition. A full version of this paper is accessible at: https://arxiv.org/pdf/2101.06453.pdf Â© 2021 IEEE.

Item Type: Conference Paper IEEE International Symposium on Information Theory - Proceedings Institute of Electrical and Electronics Engineers Inc. The copyright for this article belongs to Institute of Electrical and Electronics Engineers Inc. Information theory; Markov chains; Probability density function, Acceptance ratio; D -dimensional lattices; Integer lattice; Lipschitz conditions; Markov chain monte carlo algorithms; Metropolis Hastings; Probability densities; Proposal distribution, Probability distributions Division of Electrical Sciences > Electrical Communication Engineering 03 Dec 2021 08:41 03 Dec 2021 08:41 http://eprints.iisc.ac.in/id/eprint/70262