Algorithms for $D$-modules, integration, and generalized functions with applications to statistics

Abstract

This is an enlarged and revised version of the slides presented in a series of survey lectures given by the present author at MSJ SI 2015 in Osaka. The goal is to introduce an algorithm for computing a holonomic system of linear (ordinary or partial) differential equations for the integral of a holonomic function over the domain defined by polynomial inequalities. It applies to the cumulative function of a polynomial of several independent random variables with e.g., a normal distribution or a gamma distribution. Our method consists in Gröbner basis computation in the Weyl algebra, i.e., the ring of differential operators with polynomial coefficients. In the algorithm, generalized functions are inevitably involved even if the integrand is a usual function. Hence we need to make sure to what extent purely algebraic method of Gröbner basis applies to generalized functions which are based on real analysis.