Bayesian inference for generalized linear models via quasi-posteriors

TL;DR

This paper introduces a robust Bayesian inference method for generalized linear models using quasi-posteriors, which are resilient to model misspecification and connect to coarsened posteriors, with proven asymptotic normality and good frequentist coverage.

Contribution

It develops a novel quasi-posterior framework for Bayesian inference in GLMs, providing theoretical insights and practical tools for robustness and calibration.

Findings

Quasi-posteriors asymptotically converge to a normal distribution.

The method offers well-calibrated frequentist coverage.

Provides a new interpretation of the coarsening parameter as dispersion.

Abstract

Generalized linear models (GLMs) are routinely used for modeling relationships between a response variable and a set of covariates. The simple form of a GLM comes with easy interpretability, but also leads to concerns about model misspecification impacting inferential conclusions. A popular semi-parametric solution adopted in the frequentist literature is quasi-likelihood, which improves robustness by only requiring correct specification of the first two moments. We develop a robust approach to Bayesian inference in GLMs through quasi-posterior distributions. We show that quasi-posteriors provide a coherent generalized Bayes inference method, while also approximating so-called coarsened posteriors. In so doing, we obtain new insights into the choice of coarsening parameter. Asymptotically, the quasi-posterior converges in total variation to a normal distribution and has important connections with the loss-likelihood bootstrap posterior. We demonstrate that it is also well-calibrated in terms of frequentist coverage. Moreover, the loss-scale parameter has a clear interpretation as a dispersion, and this leads to a consolidated method of moments estimator.