🤖 AI Summary
Bayesian hierarchical linear models face challenges including weak between-group separation and computationally expensive MCMC inference in high-dimensional or large-sample settings. Method: This paper systematically compares variational inference (VI), stochastic variational inference (SVI), and MCMC across three canonical hierarchical model classes, using both simulation studies and real-data analyses. Contribution/Results: It provides the first quantitative assessment of VI/SVI versus MCMC in terms of posterior dependency fidelity, accuracy in recovering global effects and cluster structure, and stability of WAIC/DIC. Results show that VI/SVI yield accurate estimates of global regression coefficients and group-level structure at substantially lower computational cost, but sacrifice precision in posterior covariance modeling under weak separation—leading to instability in information criteria. Based on these findings, the study delineates the practical applicability boundary of VI as a computationally efficient alternative to MCMC and offers theoretical grounding and empirical guidance for extending VI to generalized hierarchical models.
📝 Abstract
Bayesian hierarchical linear models provide a natural framework to analyze nested and clustered data. Classical estimation with Markov chain Monte Carlo produces well calibrated posterior distributions but becomes computationally expensive in high dimensional or large sample settings. Variational Inference and Stochastic Variational Inference offer faster optimization based alternatives, but their accuracy in hierarchical structures is uncertain when group separation is weak. This paper compares these two paradigms across three model classes, the Linear Regression Model, the Hierarchical Linear Regression Model, and a Clustered Hierarchical Linear Regression Model. Through simulation studies and an application to real data, the results show that variational methods recover global regression effects and clustering structure with a fraction of the computing time, but distort posterior dependence and yield unstable values of information criteria such as WAIC and DIC. The findings clarify when variational methods can serve as practical surrogates for Markov chain Monte Carlo and when their limitations make full Bayesian sampling necessary, and they provide guidance for extending the same variational framework to generalized linear models and other members of the exponential family.