Power posteriors "robustify" standard Bayesian inference by raising the likelihood to a constant fractional power, effectively downweighting its influence in the calculation of the posterior. Power posteriors have been shown to be more robust to model misspecification than standard posteriors in many settings. Previous work has shown that power posteriors derived from low-dimensional, parametric locally asymptotically normal models are asymptotically normal (Bernstein-von Mises) even under model misspecification. We extend these results to show that the power posterior moments converge to those of the limiting normal distribution suggested by the Bernstein-von Mises theorem. We then use this result to show that the mean of the power posterior, a point estimator, is asymptotically equivalent to the maximum likelihood estimator.

翻译：幂后验通过将似然函数提升至恒定分数次幂，有效降低其在后验计算中的影响，从而"鲁棒化"标准贝叶斯推断。研究表明，在许多场景下，幂后验相比标准后验对模型误设具有更优的鲁棒性。已有工作证明，基于低维参数化局部渐近正态模型推导的幂后验在模型误设条件下仍具有渐近正态性（Bernstein-von Mises定理）。我们将此结论拓展至幂后验矩的收敛性，证明其收敛于Bernstein-von Mises定理所揭示的极限正态分布矩。基于该结果进一步证明，作为点估计量的幂后验均值与极大似然估计量渐近等价。