The Bayesian Infinitesimal Jackknife for Variance

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.