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Sampling according to the multivariate normal density

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2 Author(s)
Kannan, R. ; Sch. of Comput. Sci., Carnegie Mellon Univ., Pittsburgh, PA, USA ; Guangxing Li

This paper deals with the normal density of n dependent random variables. This is a function of the form: ce(-xTAx) where A is an n×n positive definite matrix, a: is the n-vector of the random variables and c is a suitable constant. The first problem we consider is the (approximate) evaluation of the integral of this function over the positive orthant ∫(x1=0)∫(x2=0)···∫(xn=0)ce(-xTAx). This problem has a long history and a substantial literature. Related to it is the problem of drawing a sample from the positive orthant with probability density (approximately) equal to ce(-xTAx). We solve both these problems here in polynomial time using rapidly mixing Markov Chains. For proving rapid convergence of the chains to their stationary distribution, we use a geometric property called the isoperimetric inequality. Such an inequality has been the subject of recent papers for general log-concave functions. We use these techniques, but the main thrust of the paper is to exploit the special property of the normal density to prove a stronger inequality than for general log-concave functions. We actually consider first the problem of drawing a sample according to the normal density with A equal to the identity matrix from a convex set K in Rn which contains the unit ball. This problem is motivated by the problem of computing the volume of a convex set in a way we explain later. Also, the methods used in the solution of this and the orthant problem are similar

Published in:

Foundations of Computer Science, 1996. Proceedings., 37th Annual Symposium on

Date of Conference:

14-16 Oct 1996