Sampling According to the Multivariate Normal Density

This paper 1 deals with the normal density of n dependent random variables. This is a function of the form : ce ?x T Ax where A is an n n positive deenite matrix, x is the n?vector of the random variables and c is a suitable constant. The rst problem we consider is the (approximate) evaluation of the integral of this function over the positive orthant Z 1 x1=0 Z 1 x2=0 ... Z 1 xn=0 ce ?x T Ax : 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 ?x T Ax. 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 rst the problem of drawing a sample according to the normal density with A equal to the identity matrix from a convex set K in R n 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.