Expectation Maximization as Message Passing - Part I: Principles and Gaussian Messages

TL;DR

This paper presents a novel perspective on expectation maximization (EM) as a message passing algorithm within factor graphs, enabling tractable Gaussian message computations especially in linear Gaussian state space models.

Contribution

It introduces a general EM message computation rule and demonstrates how EM can simplify message passing in cyclic factor graphs, particularly for Gaussian models.

Findings

EM can be viewed as message passing in factor graphs

EM simplifies message passing in cyclic graphs with Gaussian models

Tabulated EM messages for common multipliers in applications

Abstract

It is shown how expectation maximization (EM) may be viewed as a message passing algorithm in factor graphs. In particular, a general EM message computation rule is identified. As a factor graph tool, EM may be used to break cycles in a factor graph, and tractable messages may in some cases be obtained where the sum-product messages are unwieldy. As an exemplary application, the paper considers linear Gaussian state space models. Unknown coefficients in such models give rise to multipliers in the corresponding factor graph. A main attraction of EM in such cases is that it results in purely Gaussian message passing algorithms. These Gaussian EM messages are tabulated for several (scalar, vector, matrix) multipliers that frequently appear in applications.