On the Normalizing Constant of the Continuous Categorical Distribution

TL;DR

This paper investigates the normalizing constant of the continuous categorical distribution, providing insights into its numerical behavior and advancing methods to facilitate its broader application in statistics and machine learning.

Contribution

It offers a detailed characterization of the normalizing constant and introduces theoretical and methodological improvements for the continuous categorical distribution.

Findings

Normalizing constant can be expressed with elementary functions

Numerical behavior of the constant is characterized

Advances enable broader applications of the distribution

Abstract

Probability distributions supported on the simplex enjoy a wide range of applications across statistics and machine learning. Recently, a novel family of such distributions has been discovered: the continuous categorical. This family enjoys remarkable mathematical simplicity; its density function resembles that of the Dirichlet distribution, but with a normalizing constant that can be written in closed form using elementary functions only. In spite of this mathematical simplicity, our understanding of the normalizing constant remains far from complete. In this work, we characterize the numerical behavior of the normalizing constant and we present theoretical and methodological advances that can, in turn, help to enable broader applications of the continuous categorical distribution. Our code is available at https://github.com/cunningham-lab/cb_and_cc/.