I’m going to be speaking next week at Stancon 2020 about a project I’ve been trying to wrap up this summer: the Bayesian (first-order) infinitesimal jackknife. The idea is very simple: it’s a combination of the Bayesian sensitivity Theorem 1 of our Covariances, Robustness, and Variational Bayes paper with the general approach of our Swiss Army Infinitesimal Jackknife paper, applied to estimate the frequentist variance as in, say, Section 3.5 of The Influence Curve and its Role in Robust Estimation (Hampel, 1974).

To be concrete, let \(\theta \in \mathbb{R}^D\) be a real-valued parameter, \(\ell(x_n | \theta)\) be the log likelihood of datapoint \(x_n\), and \(w_n \in \mathbb{R}\) be scalar-valued weights for \(n = 1, \ldots, N\), with \(w = (w_1, \ldots, w_N)\) and \(x = (x_1, \ldots, x_N)\). We can define the weighted loglikleihood by

\[L(x | \theta, w) := \sum_{n=1}^N w_n \ell(x_n \vert \theta),\]

and form the weighted posterior

\[p(\theta \vert x, w) \propto \int \exp(L(x | \theta, w)) p(\theta).\]

Then the Bayesian empirical influence function for the posterior expectation \(\mathbb{E}[\theta | x, w]\) is given by

\[\psi_n := N \frac{d \mathbb{E}[\theta | x, w]}{ d w_n} \Biggr\vert_{w_n = 1} = N \mathrm{Cov}(\theta, \ell(x_n \vert \theta)),\]

where the final equality is given by Theorem 1 of our Covariances, Robustness, and Variational Bayes paper.

The point is just that the empirical influence function of a Bayesian posterior expectation can be computed easily from a single run of MCMC samples. And thus, so can everything that an influence function is useful for, including the infiniteismal jackknife estimate of variance:

\[\mathrm{Var}_x \left( \mathbb{E}[\theta | x] \right) \approx \frac{1}{N} \sum_{n=1}^N \psi_n^2 - \left(\frac{1}{N} \sum_{n=1}^N \psi_n \right)^2\]

wher the outermost variance is taken over the data, \(x\). At Stancon I’ll emphasize using this variance estimate as a replacement for the bootstrap when detecting or accomodating misspecification in Bayesian models, but of course you could also use it for, say approximate leave-k-out cross validation, just as in our Swiss Army Infinitesimal Jackknife paper.

The hard part about this has been proving that it works, and it’s been hard because I seem to need a version of the Bayesian central limit theorem (BCLT, also known as the Bernstein-von Mises theorem) that keeps track of the data dependence of ther residaul of the leading order expansion of a posterior expectation. Learning how the BCLT proofs work has been a slog for my feeble mind (and happens to be the reason why I’m posting about van der Vaardt). I have what I believe is a valid proof that I hope to put up on the arxiv soon.

However, you probably don’t need the proof to convince yourself that this works and try it out. I’ll show some examples at Stancon, but you can also take a look at some of the functionality I’ve implemented in our rstansensitivity package.