# Streamlined Variational Inference for Linear Mixed Models with Crossed Random Effects

arXiv: https://arxiv.org/abs/1910.01799 · HTML: https://mlsanity.com/paper/1910.01799/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Oct 2019.

## Built on by (most cited first, 4 of 4)

- [Partially factorized variational inference for high-dimensional mixed models](https://mlsanity.com/paper/2312.13148/) (2023)
- [Linear-cost unbiased posterior estimates for crossed effects and matrix factorization models via couplings](https://mlsanity.com/paper/2410.08939/) (2024)
- [Conjugate gradient methods for high-dimensional GLMMs](https://mlsanity.com/paper/2411.04729/) (2024)
- [Vector Copula Variational Inference and Dependent Block Posterior Approximations](https://mlsanity.com/paper/2503.01072/) (2025)

## Abstract (arXiv)

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.