# Scalable Bayesian computation for crossed and nested hierarchical models

arXiv: https://arxiv.org/abs/2103.10875 · HTML: https://mlsanity.com/paper/2103.10875/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Mar 2021.

## Built on by (most cited first, 9 of 9)

- [Entropy contraction of the Gibbs sampler under log-concavity](https://mlsanity.com/paper/2410.00858/) (2024)
- [Partially factorized variational inference for high-dimensional mixed models](https://mlsanity.com/paper/2312.13148/) (2023)
- [Dimension-free mixing times of Gibbs samplers for Bayesian hierarchical models](https://mlsanity.com/paper/2304.06993/) (2023)
- [Scalability of Metropolis-within-Gibbs schemes for high-dimensional Bayesian models](https://mlsanity.com/paper/2403.09416/) (2024)
- [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 develop sampling algorithms to fit Bayesian hierarchical models, the computational complexity of which scales linearly with the number of observations and the number of parameters in the model. We focus on crossed random effect and nested multilevel models, which are used ubiquitously in applied sciences. The posterior dependence in both classes is sparse: in crossed random effects models it resembles a random graph, whereas in nested multilevel models it is tree-structured. For each class we identify a framework for scalable computation, building on previous work. Methods for crossed models are based on extensions of appropriately designed collapsed Gibbs samplers, where we introduce the idea of local centering; while methods for nested models are based on sparse linear algebra and data augmentation. We provide a theoretical analysis of the proposed algorithms in some simplified settings, including a comparison with previously proposed methodologies and an average-case analysis based on random graph theory. Numerical experiments, including two challenging real data analyses on predicting electoral results and real estate prices, compare with off-the-shelf Hamiltonian Monte Carlo, displaying drastic improvement in performance.