Detailed Derivations of Small-Variance Asymptotics for some Hierarchical Bayesian Nonparametric Models

TL;DR

This paper provides detailed derivations of small-variance asymptotics for hierarchical Bayesian nonparametric models, specifically the HDP mixture models and HDP-HMM, including derivations for related partition probabilities.

Contribution

It offers comprehensive derivations of small-variance asymptotics for hierarchical Dirichlet process models, enhancing understanding of their behavior and properties.

Findings

Derivations for HDP mixture models and HDP-HMM

Probabilities of CRP and CRF partitions detailed

Clarifies the mathematical foundations of small-variance asymptotics

Abstract

In this note we provide detailed derivations of two versions of small-variance asymptotics for hierarchical Dirichlet process (HDP) mixture models and the HDP hidden Markov model (HDP-HMM, a.k.a. the infinite HMM). We include derivations for the probabilities of certain CRP and CRF partitions, which are of more general interest.