Neural Estimation of Summary Statistics for Simulation-Based Inference with Non-Asymptotic Guarantees

Abstract: Simulation-based inference (SBI) enables Bayesian inference when a simulator is available but the likelihood is intractable. Modern SBI methods train neural networks on simulated data to obtain an amortized approximation of the posterior, making inference for new observations fast. A central challenge in SBI is to extract informative, low-dimensional summary statistics; without them, performance can degrade rapidly as the observation dimension grows. We therefore highlight summary learning as a critical component of SBI and propose a general framework that jointly learns a neural-network-parameterized summary and an amortized posterior estimator. We further introduce a local refinement strategy that adapts the amortized estimator to a specific observation, thereby reducing the amortization gap. Both amortized training and local refinement accommodate implicit posterior estimators without tractable densities. On the theoretical side, we analyze the finite-sample behavior of the joint learning procedure through the lens of nonparametric conditional distribution estimation, and show on representative examples how low-dimensional summaries can mitigate the classical curse of dimensionality. Empirically, our results support this motivation and demonstrate favorable performance relative to existing methods.