Streamlined Variational Inference for Linear Mixed Models with Crossed Random Effects

TL;DR

This paper develops and compares streamlined variational inference algorithms for linear mixed models with crossed random effects, balancing accuracy and computational efficiency for large-scale problems.

Contribution

It introduces three variational inference strategies with detailed algorithms and empirical evaluations for fitting complex linear mixed models.

Findings

High accuracy with relaxed mean field assumptions

Sparse algorithms improve computational speed

Trade-offs between accuracy and efficiency are characterized

Abstract

We derive streamlined mean field variational Bayes algorithms for fitting linear mixed models with crossed random effects. In the most general situation, where the dimensions of the crossed groups are arbitrarily large, streamlining is hindered by lack of sparseness in the underlying least squares system. Because of this fact we also consider a hierarchy of relaxations of the mean field product restriction. The least stringent product restriction delivers a high degree of inferential accuracy. However, this accuracy must be mitigated against its higher storage and computing demands. Faster sparse storage and computing alternatives are also provided, but come with the price of diminished inferential accuracy. This article provides full algorithmic details of three variational inference strategies, presents detailed empirical results on their pros and cons and, thus, guides the users on their choice of variational inference approach depending on the problem size and computing resources.