Bayesian inference for generalized linear models via quasi-posteriors

Abstract

Generalized linear models are routinely used for modelling relationships between a response variable and a set of covariates. The simple form of a generalized linear model comes with easy interpretability, but also leads to concerns about model misspecification impacting inferential conclusions. A popular semiparametric solution adopted in the frequentist literature is quasilikelihood, which improves robustness by only requiring correct specification of the first two moments. We develop a robust approach to Bayesian inference in generalized linear models 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.

Cite this paper

@article{agnoletto2025bayesian,
  title = {{Bayesian inference for generalized linear models via quasi-posteriors}},
  author = {Agnoletto, Davide and Rigon, Tommaso and Dunson, David B.},
  journal = {Biometrika},
  volume = {112},
  number = {2},
  pages = {asaf022},
  year = {2025},
  doi = {10.1093/biomet/asaf022}
}

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