Bayesian inference is optimal when the statistical model is well-specified, while outside this setting Bayesian inference can catastrophically fail; accordingly a wealth of post-Bayesian methodologies have been proposed. Predictively oriented (PrO) approaches lift the statistical model $P_θ$ to an (infinite) mixture model $\int P_θ\; \mathrm{d}Q(θ)$ and fit this predictive distribution via minimising an entropy-regularised objective functional. In the well-specified setting one expects the mixing distribution $Q$ to concentrate around the true data-generating parameter in the large data limit, while such singular concentration will typically not be observed if the model is misspecified. Our contribution is to demonstrate that one can empirically detect model misspecification by comparing the standard Bayesian posterior to the PrO `posterior' $Q$. To operationalise this, we present an efficient numerical algorithm based on variational gradient descent. A simulation study, and a more detailed case study involving a Bayesian inverse problem in seismology, confirm that model misspecification can be automatically detected using this framework.

翻译：当统计模型设定正确时，贝叶斯推断是最优的；而在此设定之外，贝叶斯推断可能灾难性地失效。为此，学界已提出大量后贝叶斯方法论。面向预测（PrO）方法将统计模型 $P_θ$ 提升为（无限）混合模型 $\int P_θ\; \mathrm{d}Q(θ)$，并通过最小化熵正则化目标泛函来拟合该预测分布。在模型设定正确的情况下，当数据量趋于无穷时，混合分布 $Q$ 会集中收敛于真实数据生成参数；而当模型存在误设时，通常不会观测到此类奇异收敛现象。我们的贡献在于证明：通过比较标准贝叶斯后验与 PrO"后验" $Q$，可以经验性地检测模型误设。为实现这一检测，我们提出了一种基于变分梯度下降的高效数值算法。模拟实验以及涉及地震学贝叶斯反问题的详细案例研究表明，该框架能够自动检测模型误设。