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常数分数次幂来“稳健化”标准贝叶斯推断，有效降低了似然在后验计算中的影响。研究表明，幂后验在许多场景下比标准后验对模型误设更具稳健性。已有工作证明，即使在模型误设条件下，基于低维参数局部渐近正态模型导出的幂后验仍具有渐近正态性（Bernstein-von Mises定理）。我们进一步推广该结论，证明幂后验矩收敛至Bernstein-von Mises定理所给出的极限正态分布的矩。基于此结果，我们论证了作为点估计量的幂后验均值在渐近意义上与最大似然估计等价。