From the Expectation Maximisation Algorithm to Autoencoded Variational Bayes

TL;DR

This paper provides a comprehensive tutorial linking the classical EM algorithm to modern autoencoded variational Bayes and variational autoencoders, clarifying key concepts and their evolution.

Contribution

It offers a detailed, unified explanation of EM, variational Bayesian inference, and autoencoders, highlighting their connections and clarifying the meaning of latent variables.

Findings

Clarifies the relationship between EM and variational inference.

Explains the reparametrisation trick in detail.

Provides numerical examples illustrating the algorithms.

Abstract

Although the expectation maximisation (EM) algorithm was introduced in 1970, it remains somewhat inaccessible to machine learning practitioners due to its obscure notation, terse proofs and lack of concrete links to modern machine learning techniques like autoencoded variational Bayes. This has resulted in gaps in the AI literature concerning the meaning of such concepts like "latent variables" and "variational lower bound," which are frequently used but often not clearly explained. The roots of these ideas lie in the EM algorithm. We first give a tutorial presentation of the EM algorithm for estimating the parameters of a $K$-component mixture density. The Gaussian mixture case is presented in detail using $K$-ary scalar hidden (or latent) variables rather than the more traditional binary valued $K$-dimenional vectors. This presentation is motivated by mixture modelling from the target tracking literature. In a similar style to Bishop's 2009 book, we present variational Bayesian inference as a generalised EM algorithm stemming from the variational (or evidential) lower bound, as well as the technique of mean field approximation (or product density transform). We continue the evolution from EM to variational autoencoders, developed by Kingma & Welling in 2014. In so doing, we establish clear links between the EM algorithm and its variational counterparts, hence clarifying the meaning of "latent variables." We provide a detailed coverage of the "reparametrisation trick" and focus on how the AEVB differs from conventional variational Bayesian inference. Throughout the tutorial, consistent notational conventions are used. This unifies the narrative and clarifies the concepts. Some numerical examples are given to further illustrate the algorithms.