Expectation Propagation on the Maximum of Correlated Normal Variables

TL;DR

This paper develops a method to approximate the distribution of the maximum of correlated Gaussian variables using Expectation Propagation, enabling efficient inference in problems involving maxima of Gaussian sets.

Contribution

It derives the first two posterior moments of the maximum of two correlated Gaussians and introduces a heuristic for approximate inference over multiple Gaussian variables.

Findings

Derived posterior moments for the maximum of two correlated Gaussians.

Proposed a heuristic for approximate inference on maxima of Gaussian sets.

Enabled Expectation Propagation-based inference for maximum-related problems.

Abstract

Many inference problems involving questions of optimality ask for the maximum or the minimum of a finite set of unknown quantities. This technical report derives the first two posterior moments of the maximum of two correlated Gaussian variables and the first two posterior moments of the two generating variables (corresponding to Gaussian approximations minimizing relative entropy). It is shown how this can be used to build a heuristic approximation to the maximum relationship over a finite set of Gaussian variables, allowing approximate inference by Expectation Propagation on such quantities.