A Rejection Technique for Sampling from Log-Concave Multivariate Distributions

Josef Leydold


Different universal methods (also called automatic or black-box methods) have been suggested to sample from univariate log-concave distributions. The description of a suitable universal generator for multivariate distributions in arbitrary dimensions has not been published up to now. The new algorithm is based on the method of transformed density rejection. To construct a hat function for the rejection algorithm the multivariate density is tranformed by a proper transformation T into a concave function (in the case of log-concave density T(x) = log(x).) Then it is possible to construct a dominating function by taking the minimum of several tangent hyperplanes which are transformed back by $T^{-1}$ into the original scale. The domains of different pieces of the hat function are polyhedra in the multivariate case. Although this method can be shown to work, it is too slow and complicated in higher dimensions. In this paper we split the $R^n$ into simple cones. The hat function is constructed piecewise on each of the cones by tangent hyperplanes. The resulting function is not continuous any more and the rejection constant is bounded from below but the setup and the generation remains quite fast in higher dimensions, e.g. n=8. The paper describes the details how this main idea can be used to construct algorithm TDRMV that generates random tuples from multivariate log-concave distribution with a computable density. Although the developed algorithm is not a real black box method it is adjustable for a large class of log-concave densities.

CR Categories and Subject Descriptors: G.3 [Probability and Statistics]: Random number generation

General Terms: Algorithms

Key Words: Rejection method, multivariate log-concave distributions, universal method

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© ACM, (1998). This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in Trans. Model. Comput. Simul. 8(3), 254-280. http://doi.acm.org/10.1145/290274.290287