We provide a general solution to a fundamental open problem in Bayesian inference, namely poor uncertainty quantification, from a frequency standpoint, of Bayesian methods in misspecified models. While existing solutions are based on explicit Gaussian approximations of the posterior, or computationally onerous post-processing procedures, we demonstrate that correct uncertainty quantification can be achieved by replacing the usual posterior with an intuitive approximate posterior. Critically, our solution is applicable to likelihood-based, and generalized, posteriors as well as cases where the likelihood is intractable and must be estimated. We formally demonstrate the reliable uncertainty quantification of our proposed approach, and show that valid uncertainty quantification is not an asymptotic result but occurs even in small samples. We illustrate this approach through a range of examples, including linear, and generalized, mixed effects models.

翻译：我们针对贝叶斯推断中一个基础性未解决问题——即从频率学派角度看，贝叶斯方法在误设模型中不确定性量化不佳——提出了通用解决方案。现有方法基于后验的显式高斯近似或计算繁重的后处理程序，而我们的研究表明，通过用直观的近似后验替代常规后验即可实现正确的不确定性量化。关键在于，本方案适用于基于似然函数和广义后验，以及似然函数难以处理而需估计的情形。我们正式论证了所提方法的不确定性量化可靠性，并表明有效的不确定性量化并非渐进结果，在小样本条件下同样成立。通过包括线性模型、广义线性混合效应模型在内的一系列实例展示了该方法的有效性。