On the Normalizing Constant of the Continuous Categorical Distribution
This work addresses a technical gap in probability theory for researchers and practitioners using simplex-supported distributions, but it is incremental as it builds on a recently discovered family.
The paper tackles the incomplete understanding of the normalizing constant in the continuous categorical distribution, presenting theoretical and methodological advances to characterize its numerical behavior and enable broader applications.
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/.