Bayesian Generalization Error of Poisson Mixture and Simplex Vandermonde Matrix Type Singularity

TL;DR

This paper derives the Bayesian generalization error for Poisson mixture models by calculating the Real Log Canonical Threshold (RLCT) associated with a specific singularity type, advancing theoretical understanding.

Contribution

It provides the first theoretical analysis of the RLCT for Poisson mixtures with simplex Vandermonde matrix singularities, linking singularity theory to Bayesian generalization error.

Findings

RLCT of the simplex Vandermonde matrix singularity is derived

The RLCT matches that of Poisson mixture models in general cases

Provides a theoretical foundation for understanding Bayesian generalization in singular models

Abstract

A Poisson mixture is one of the practically important models in computer science, biology, and sociology. However, the theoretical property has not been studied because the posterior distribution can not be approximated by any normal distribution. Such a model is called singular and it is known that Real Log Canonical Threshold (RLCT) is equal to the coefficient of the asymptotically main term of the Bayesian generalization error. In this paper, we derive RLCT of a simplex Vandermonde matrix type singularity which is equal to that of a Poisson mixture in general cases.