Posterior accuracy and calibration under misspecification in Bayesian generalized linear models

TL;DR

This study investigates how different likelihoods and link functions in Bayesian generalized linear models affect posterior accuracy and calibration under misspecification, providing practical guidance for robust model selection.

Contribution

It offers a comprehensive simulation analysis of likelihood and link robustness in Bayesian GLMs, including calibration and uncertainty metrics, which was lacking in prior research.

Findings

Certain likelihoods and links are highly robust to misspecification.

Normal likelihood with identity link often achieves good calibration.

Robust choices can perform nearly as well as true models.

Abstract

Generalized linear models (GLMs) are popular for data-analysis in almost all quantitative sciences, but the choice of likelihood family and link function is often difficult. This motivates the search for likelihoods and links that minimize the impact of potential misspecification. We perform a large-scale simulation study on double-bounded and lower-bounded response data where we systematically vary both true and assumed likelihoods and links. In contrast to previous studies, we also study posterior calibration and uncertainty metrics in addition to point-estimate accuracy. Our results indicate that certain likelihoods and links can be remarkably robust to misspecification, performing almost on par with their respective true counterparts. Additionally, normal likelihood models with identity link (i.e., linear regression) often achieve calibration comparable to the more structurally faithful alternatives, at least in the studied scenarios. On the basis of our findings, we provide practical suggestions for robust likelihood and link choices in GLMs.