The frequentist variability of Bayesian posterior expectations can provide meaningful measures of uncertainty even when models are misspecified. Classical methods to asymptotically approximate the frequentist covariance of Bayesian estimators such as the Laplace approximation and the nonparametric bootstrap can be practically inconvenient, since the Laplace approximation may require an intractable integral to compute the marginal log posterior, and the bootstrap requires computing the posterior for many different bootstrap datasets. We develop and explore the infinitesimal jackknife (IJ), an alternative method for computing asymptotic frequentist covariance of smooth functionals of exchangeable data, which is based on the "influence function" of robust statistics. We show that the influence function for posterior expectations has the form of a simple posterior covariance, and that the IJ covariance estimate is, in turn, easily computed from a single set of posterior samples. Under conditions similar to those required for a Bayesian central limit theorem to apply, we prove that the corresponding IJ covariance estimate is asymptotically equivalent to the Laplace approximation and the bootstrap. In the presence of nuisance parameters that may not obey a central limit theorem, we argue using a von Mises expansion that the IJ covariance is inconsistent, but can remain a good approximation to the limiting frequentist variance. We demonstrate the accuracy and computational benefits of the IJ covariance estimates with simulated and real-world experiments.

翻译：即使模型存在误设，贝叶斯后验期望的频率派变异性仍能提供有意义的度量。经典方法（如拉普拉斯近似和非参数自助法）虽能渐近逼近贝叶斯估计量的频率协方差，但实际应用存在局限：拉普拉斯近似需通过复杂积分计算边缘对数后验，而自助法则需对大量重抽样数据集重复计算后验分布。本文提出并探讨了无穷小刀切法——一种基于鲁棒统计学中“影响函数”的新方法，用于计算可交换数据光滑泛函的渐近频率协方差。我们证明后验期望的影响函数可表示为简单的后验协方差形式，进而可通过单组后验样本便捷计算IJ协方差估计。在满足贝叶斯中心极限定理的条件下，我们证明IJ协方差估计与拉普拉斯近似及自助法具有渐近等价性。当存在不满足中心极限定理的冗余参数时，通过冯·米塞斯展开论证表明：虽然IJ协方差估计存在不一致性，但仍能较好逼近极限频率方差。最后，我们通过模拟实验和真实数据实验验证了IJ协方差估计的准确性与计算优势。