Mean Field Variational Bayesian Inference and Statistical Mechanics of Gaussian Mixture Model

TL;DR

This paper analyzes the theoretical foundations of Mean Field Variational Bayesian Inference for Gaussian Mixture Models, linking it to statistical mechanics concepts like partition functions and free energy.

Contribution

It provides a rigorous understanding of uncertainty quantification in MFVBI for GMM, connecting it to the Curie-Weiss model from statistical mechanics.

Findings

GMM can be viewed as a generalization of the Curie-Weiss model.

Partition function and free energy naturally emerge in the analysis.

Offers a definitive theoretical framework for MFVBI in GMM.

Abstract

One of the main modeling in many data science applications is the Gaussian Mixture Model (GMM), and Mean Field Variational Bayesian Inference (MFVBI) is classically used for approximate fast computation. In this paper, we provide a definitive answer to the fundamental inquiry about the uncertainty quantification of the MFVBI applied to the GMM. It turns out that GMM can be considered as a generalization of Curie--Weiss model in statistical mechanics. The standard quantities like partition function and free energy appear naturally in the process of our analysis.