基于准后验分布的广义线性模型贝叶斯推断
Bayesian inference for generalized linear models via quasi-posteriors
AI总结：
针对广义线性模型的模型误设问题，提出基于准后验分布的稳健贝叶斯推断方法，证明其自洽性与渐近性质，验证频率校准性并给出统一矩估计方案。
摘要：
广义线性模型（GLMs）常被用于建模响应变量与一组协变量之间的关系。GLM形式简洁，易于解释，但也引发了模型误设会影响推断结论的担忧。频率学界常用的一种半参数解决方案是准似然法，该方法仅要求正确设定前两阶矩，从而提升了稳健性。我们通过准后验分布提出了一种稳健的GLM贝叶斯推断方法。我们证明，准后验分布提供了一种自洽的广义贝叶斯推断方法，同时还能近似所谓的粗化后验分布，借此我们对粗化参数的选择获得了新的认知。在渐近性质上，准后验分布在全变差意义下收敛于正态分布，且与损失-似然自助后验分布存在重要关联。我们证明其在频率覆盖度方面也具备良好的校准性。此外，损失尺度参数可被明确解释为离散参数，由此衍生出一种统一的矩估计方法。
英文摘要：
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.
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