# Scalable inference for crossed random effects models

arXiv: https://arxiv.org/abs/1803.09460 · HTML: https://mlsanity.com/paper/1803.09460/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Mar 2018.

## Built on by (most cited first, 8 of 8)

- [Partially factorized variational inference for high-dimensional mixed models](https://mlsanity.com/paper/2312.13148/) (2023)
- [Scalability of Metropolis-within-Gibbs schemes for high-dimensional Bayesian models](https://mlsanity.com/paper/2403.09416/) (2024)
- [Quasi-Newton Quasi-Monte Carlo for variational Bayes](https://mlsanity.com/paper/2104.02865/) (2021)
- [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)
- [Scalable and Calibrated Sampling for Bayesian Generalized Linear Mixed Model via Stochastic Gradient Markov Chain Monte Carlo](https://mlsanity.com/paper/2403.03007/) (2024)
- [Hamiltonian Monte Carlo Inference of Marginalized Linear Mixed-Effects Models](https://mlsanity.com/paper/2410.24079/) (2024)
- [Scalable Krylov Subspace Methods for Generalized Mixed-Effects Models with Crossed Random Effects](https://mlsanity.com/paper/2505.09552/) (2025)

## Abstract (arXiv)

We analyze the complexity of Gibbs samplers for inference in crossed random effect models used in modern analysis of variance. We demonstrate that for certain designs the plain vanilla Gibbs sampler is not scalable, in the sense that its complexity is worse than proportional to the number of parameters and data. We thus propose a simple modification leading to a collapsed Gibbs sampler that is provably scalable. Although our theory requires some balancedness assumptions on the data designs, we demonstrate in simulated and real datasets that the rates it predicts match remarkably the correct rates in cases where the assumptions are violated. We also show that the collapsed Gibbs sampler, extended to sample further unknown hyperparameters, outperforms significantly alternative state of the art algorithms.