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.

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