Neural Variational Cut Posteriors without Upstream Data

Abstract

In many applications, one must propagate parameter uncertainty from an earlier (upstream) analysis, available as samples, to subsequent (downstream) analyses without feedback. This problem is called cutting feedback or cut-Bayes, and the cut-posterior, the optimal posterior preserving information-flow constraints, is well characterized. However, sampling from it (e.g., via nested MCMC) is computationally intensive, while existing variational inference methods for cut-Bayes require access to upstream data and model, often unavailable. We propose a modular and provably accurate cut-Bayes approach requiring no access to upstream data or model. We leverage the characterization of the cut-posterior as the minimizer of the expected downstream conditional Kullback-Leibler divergence over the upstream posterior, replacing the expectation with the sample average over upstream draws. Our method, NeVI-Cut (neural variational inference for cut-Bayes), employs conditional normalizing flows as the variational family for downstream parameters. We provide fixed-data convergence rates of NeVI-Cut in terms of the richness of neural architecture and complexity of the cut-posterior. We establish, to our knowledge, first results on uniform Kullback-Leibler approximation rates of conditional distributions by common flow classes, yielding widely applicable fixed-data error rates for variational flows. A stochastic algorithm implements NeVI-Cut efficiently, and we demonstrate its speed and accuracy on multiple applications.