Fast Methods for Posterior Inference of Two-Group Normal-Normal Models

Area of Science:

• Statistical modeling
• Computational statistics

Background:

• Bayesian linear regression models are widely used.
• Hierarchical and random effects models present computational challenges.
• Markov chain Monte Carlo (MCMC) methods are common but can be slow and difficult to tune.

Purpose of the Study:

• To develop novel algorithms for evaluating posterior moments in Bayesian linear regression.
• To provide efficient computational methods for hierarchical mixed effects and random effects models.
• To reduce computational cost and improve tuning of Bayesian models.

Main Methods:

• Analytical marginalization of regression coefficients.
• Numerical integration of low-dimensional densities.
• Eigendecomposition as the dominant computational cost.

Main Results:

• Algorithms are applicable to hierarchical models with partial pooling.
• Demonstrated performance on U.S. opinion polls and COVID-19 outbreak data.
• Significantly reduced run times compared to state-of-the-art MCMC algorithms.

Conclusions:

• The proposed algorithms offer a computationally efficient alternative for Bayesian linear regression.
• These methods are particularly beneficial for complex hierarchical and random effects models.
• The approach simplifies model tuning and reduces computational burden.