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Sampling from Log-Concave Distributions (1994)
| Venue: | Ann. Appl. Prob |
| Citations: | 36 - 3 self |
BibTeX
@ARTICLE{Frieze94samplingfrom,
author = {Alan Frieze and Ravi Kannan and Nick Polson},
title = {Sampling from Log-Concave Distributions},
journal = {Ann. Appl. Prob},
year = {1994},
volume = {4},
pages = {812--837}
}
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Abstract
This paper is concerned with the efficient sampling of random points from n, where the underlying density F is log-concave (i.e., log F is concave). This is a natural restriction which is satisfied by many common distributions, for example, the multivariate normal. The algorithm we use generates a sample path from a Markov chain whose stationary distribution is (close to) F. The algorithm falls into the class of Metropolis algorithms. It has applications to the problem of computing the volume of convex bodies and in statistics. Recent statistics literature has focused on the many applications of Markov chain Monte Carlo algorithms (see [11]). However, theoretical bounds on the convergence rate of such algorithms have been limited. Using recent developments in the theory of rapidly mixing Markov chains, in particular the notion of conductance [6, 10], Applegate and Kannan [2] proved a bound on the convergence rate of the chain considered in this paper. These theoretical bounds are too large to be useful in practice. In this paper, we prove tighter bounds using an approach related to the classical Poincar inequalities instead of conductance. These bounds are small enough to be useful in practice and have already found application in the generation of random contingency tables [4]. Instead of sampling from the continuum of points in n, we discretize the problem by assuming that n is divided into a set of hypercubes n of side ( is a given small positive real number) and the problem is to choose one of these cubes each with probability proportional to the integral of F over the cube. (If necessary, a sample from the continuum can then be picked by standard rejection sampling techniques from the cube chosen; we omit details of this.) Second, we assume that we have a compact ...
Keyphrases
log-concave distribution convergence rate markov chain theoretical bound random point metropolis algorithm natural restriction many application underlying density recent statistic literature classical poincar inequality recent development standard rejection markov chain monte carlo algorithm stationary distribution random contingency table efficient sampling many common distribution convex body sample path probability proportional small positive real number
