# Scalable and Calibrated Sampling for Bayesian Generalized Linear Mixed Model via Stochastic Gradient Markov Chain Monte Carlo

arXiv: https://arxiv.org/abs/2403.03007 · HTML: https://mlsanity.com/paper/2403.03007/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 53 references that are in the map.
- First posted on arXiv 5 Mar 2024.

## Builds on (most cited first, 16 of 16)

- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networks](https://mlsanity.com/paper/1512.07666/) (2015)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Consistency and fluctuations for stochastic gradient Langevin dynamics](https://mlsanity.com/paper/1409.0578/) (2014)
- [The promises and pitfalls of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1811.10072/) (2018)
- [Stochastic gradient Markov chain Monte Carlo](https://mlsanity.com/paper/1907.06986/) (2019)
- [Control Variates for Stochastic Gradient MCMC](https://mlsanity.com/paper/1706.05439/) (2017)
- [Bayesian Computing with INLA: A Review](https://mlsanity.com/paper/1604.00860/) (2016)
- [Conjugate Bayes for probit regression via unified skew-normal distributions](https://mlsanity.com/paper/1802.09565/) (2018)
- [Scalable Bayesian computation for crossed and nested hierarchical models](https://mlsanity.com/paper/2103.10875/) (2021)
- [Scalable inference for crossed random effects models](https://mlsanity.com/paper/1803.09460/) (2018)
- [Backfitting for large scale crossed random effects regressions](https://mlsanity.com/paper/2007.10612/) (2020)
- [Unbiased Estimation of the Gradient of the Log-Likelihood in Inverse Problems](https://mlsanity.com/paper/2003.04896/) (2020)
- [Consistent and Scalable Composite Likelihood Estimation of Probit Models with Crossed Random Effects](https://mlsanity.com/paper/2308.15681/) (2023)

## Built on by (most cited first, 1 of 1)

- [Safe, Scalable, and Accurate Bayes Posterior Sampling for Large-Data Generalized Linear Mixed Models](https://mlsanity.com/paper/2604.26029/) (2026)

## Abstract (arXiv)

Generalized linear mixed models (GLMMs) are widely used for analyzing correlated data, particularly in large-scale biomedical and social science applications. Scalable Bayesian inference for GLMMs is challenging due to an intractable marginal likelihood and a high computational cost incurred by conventional Markov chain Monte Carlo (MCMC) methods. We develop a stochastic gradient MCMC (SGMCMC) algorithm tailored to GLMMs that enables accurate posterior inference in the large-sample regime. Our approach uses Fisher's identity to construct a (biased) Monte Carlo estimator of the gradient of the marginal log-likelihood, making SGMCMC feasible when direct gradient computation is impossible. We analyze the additional variability, introduced by both data subsampling and gradient approximation, to derive a post-hoc covariance correction that yields properly calibrated posterior uncertainty. We show through simulated studies that the proposed method provides accurate posterior means and variances in settings with a large number of groups, outperforming existing approaches, including control variate methods. We further demonstrate the method's practical utility in an analysis of electronic health records data, where accounting for variance inflation materially changes scientific conclusions.