Scalable Bayesian inference for the generalized linear mixed model (2403.03007v2)

Abstract: The generalized linear mixed model (GLMM) is a popular statistical approach for handling correlated data, and is used extensively in applications areas where big data is common, including biomedical data settings. The focus of this paper is scalable statistical inference for the GLMM, where we define statistical inference as: (i) estimation of population parameters, and (ii) evaluation of scientific hypotheses in the presence of uncertainty. AI learning algorithms excel at scalable statistical estimation, but rarely include uncertainty quantification. In contrast, Bayesian inference provides full statistical inference, since uncertainty quantification results automatically from the posterior distribution. Unfortunately, Bayesian inference algorithms, including Markov Chain Monte Carlo (MCMC), become computationally intractable in big data settings. In this paper, we introduce a statistical inference algorithm at the intersection of AI and Bayesian inference, that leverages the scalability of modern AI algorithms with guaranteed uncertainty quantification that accompanies Bayesian inference. Our algorithm is an extension of stochastic gradient MCMC with novel contributions that address the treatment of correlated data (i.e., intractable marginal likelihood) and proper posterior variance estimation. Through theoretical and empirical results we establish our algorithm's statistical inference properties, and apply the method in a large electronic health records database.

• The paper introduces a novel Monte Carlo estimator to compute the marginal log-likelihood gradient for scalable Bayesian inference in GLMMs.
• The paper develops an analytical variance correction framework to mitigate the inflated posterior variance common in SGLD applications to dependent data.
• The paper validates the approach with theoretical bounds and extensive simulations, demonstrating improved inference accuracy across various GLMM specifications.

Scalable Bayesian Inference for the Generalized Linear Mixed Model

The paper explores the problem of scalable Bayesian inference within the framework of Generalized Linear Mixed Models (GLMMs), which are extensively utilized for handling correlated data in settings such as biomedical applications involving big data. The challenge addressed here is the intractable nature of traditional Bayesian inference, particularly with methods like Markov Chain Monte Carlo (MCMC), when faced with the computational demands of big data. This work proposes an algorithm that utilizes stochastic gradient MCMC (SGMCMC) approaches, marrying the scalability of AI-driven estimation techniques with the comprehensive uncertainty quantification native to Bayesian methods.

Core Contributions

1. Monte Carlo Estimator for Marginal Log-Likelihood Gradient: The authors introduce a Monte Carlo approach to estimate the gradient of the marginal log-likelihood, enabling the application of SGMCMC to GLMMs. This approach utilizes Fisher's identity, expressing the gradient as an expectation with respect to the posterior distribution of subject-specific parameters, thus making Bayesian inference computationally feasible in the GLMM setting.
2. Correcting Posterior Variance Inflation: A significant issue with naive applications of SGLD (Stochastic Gradient Langevin Dynamics) is the inflated variance in posterior estimates, particularly problematic in dependent data scenarios like GLMMs. The paper provides an analytical framework to correct this variance inflation, adjusting the covariance estimation through a Lyapunov equation derived from the properties of the noise injected by the minibatch stochastic gradient and the resultant scaling from the large dataset regime.

Theoretical and Empirical Validation

The authors present theoretical arguments supported by empirical analysis that validate the proposed methodology. The central theorem in the work provides bounds on the covariance of the error introduced by stochastic approximations in the limit where data size grows to infinity, ensuring the bias correction applied to the posterior samples is asymptotically correct.

Empirical validation is conducted through extensive simulations under various GLMM specifications, including Gaussian, Poisson, and Bernoulli distributions, both in scenarios with known and unknown variance components. The corrected algorithm consistently provides accurate inference, unlike the uncorrected version, which suffers from posterior variance inflation.

Real-World Application

The practical utility of the algorithm is demonstrated through an analysis of a large electronic health records dataset, concerning psychiatric distress in ophthalmic patients. The paper illustrates the algorithm's ability to discern relevant patient characteristics impacting distress probability, with the covariance correction improving the reliability of statistical significance tests.

Implications and Future Work

The scalable inference framework proposed here has profound implications for extending Bayesian methods to the realms traditionally dominated by frequentist approaches due to computational constraints. The method facilitates hypothesis testing and uncertainty quantification in predictive tasks over large, complex datasets.

Future directions suggested by the work include adapting the solution to high-dimensional predictor spaces, momentum-based SGMCMC variants, and further exploration into federated learning scenarios. Moreover, addressing model misspecification in real dataset applications remains an open research area, particularly its effect on the adequacy of the proposed variance correction.

Overall, this paper significantly contributes to the toolkit for statistical inference in big data settings, preserving the strengths of Bayesian methods while ensuring computational viability.