Estimation of exponential-polynomial distribution by holonomic gradient descent

TL;DR

This paper introduces a holonomic gradient descent method for efficiently estimating parameters of exponential-polynomial distributions, including extensions to multivariate cases, despite the lack of closed-form normalizing constants.

Contribution

It demonstrates the application of holonomic gradient descent for MLE of exponential-polynomial distributions and extends the approach to multivariate distributions on the entire real line.

Findings

MLE can be computed efficiently using holonomic gradient descent

Extensions to multivariate distributions on positive orthant

Method handles distributions without closed-form normalizing constants

Abstract

We study holonomic gradient decent for maximum likelihood estimation of exponential-polynomial distribution, whose density is the exponential function of a polynomial in the random variable. We first consider the case that the support of the distribution is the set of positive reals. We show that the maximum likelihood estimate (MLE) can be easily computed by the holonomic gradient descent, even though the normalizing constant of this family does not have a closed-form expression and discuss determination of the degree of the polynomial based on the score test statistic. Then we present extensions to the whole real line and to the bivariate distribution on the positive orthant.