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.

翻译：广义线性模型（GLMs）常用于建模响应变量与一组协变量之间的关系。GLM的简洁形式虽然便于解释，但也引发了关于模型错误设定影响推断结论的担忧。频率学派文献中常用的一种半参数解决方案是拟似然法，该方法通过仅要求正确指定前两阶矩来提高鲁棒性。我们通过拟后验分布发展了GLM中贝叶斯推断的鲁棒方法。研究表明，拟后验不仅提供了一种连贯的广义贝叶斯推断方法，同时还能近似所谓的粗化后验。在此过程中，我们获得了关于粗化参数选择的新见解。渐近地，拟后验在总变差距离下收敛到正态分布，并与损失-似然自举后验存在重要联系。我们还证明其在频率覆盖方面具有良好的校准性能。此外，损失尺度参数作为离散度具有清晰的解释，这引出了统一的矩估计方法。