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 heuristically that the IJ covariance 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：贝叶斯后验期望的频率派变异性即使在模型设定错误时也能提供有意义的置信度度量。经典方法如拉普拉斯近似和非参数自助法虽能渐近近似贝叶斯估计量的频率协方差，但在实践中存在不便：拉普拉斯近似可能需要计算难以处理的边际对数后验积分，而自助法则需为多个不同自助数据集重复计算后验分布。我们探索并发展了无穷小刀切法——一种基于稳健统计中"影响函数"的替代方法，用于计算可交换数据平滑泛函的渐近频率协方差。研究表明，后验期望的影响函数具有简单后验协方差形式，且无穷小刀切协方差估计可通过单组后验样本轻松计算。在满足贝叶斯中心极限定理适用条件的相似前提下，我们证明了无穷小刀切协方差估计与拉普拉斯近似及自助法具有渐近等价性。对于可能存在非中心极限定理行为的冗余参数，我们通过启发式论证指出无穷小刀切协方差仍能良好逼近极限频率方差。通过模拟实验与真实数据实验，我们验证了无穷小刀切协方差估计的准确性与计算效率优势。