Gaussian-Dirichlet Posterior Dominance in Sequential Learning

TL;DR

This paper demonstrates that in sequential learning with categorical data, the Dirichlet posterior's mean second-order stochastically dominates the Gaussian posterior's mean, offering a new analytical tool.

Contribution

It establishes a formal ordering between Dirichlet and Gaussian posteriors in sequential learning, enhancing analysis of categorical outcome models.

Findings

Dirichlet posterior mean second-order stochastically dominates Gaussian posterior mean.

Results apply when conditioned on at least two observations.

Provides a new analytical tool for sequential learning with categorical data.

Abstract

We consider the problem of sequential learning from categorical observations bounded in [0,1]. We establish an ordering between the Dirichlet posterior over categorical outcomes and a Gaussian posterior under observations with N(0,1) noise. We establish that, conditioned upon identical data with at least two observations, the posterior mean of the categorical distribution will always second-order stochastically dominate the posterior mean of the Gaussian distribution. These results provide a useful tool for the analysis of sequential learning under categorical outcomes.