Neural Posterior Estimation via Hierarchical Bayesian Distillation

Abstract

Solving inverse problems for spectral data is fundamentally hindered by data scarcity, ill-posedness, and the strict requirement for accurate uncertainty quantification (UQ). Addressing these challenges in the context of X-ray diffraction (XRD), an essential technique for materials discovery and structural health monitoring, we propose an amortized neural posterior estimation framework. Our approach distills a computationally expensive, physics-informed Bayesian teacher into a highly efficient conditional normalizing flow (CNF) student. By enforcing crystallographic constraints within our base generative model, we construct a representative synthetic training corpus to optimize the CNF, effectively bypassing the bottleneck of limited real-world data. We extensively validate the neural posterior's fidelity against the Bayesian teacher using posterior predictive checks and Wasserstein distances. Finally, we demonstrate the practical utility of this approach on a real-world structural damage detection task. Evaluated in a leave-one-group-out cross-validation (LOGOCV) setup, our amortized posteriors maintain the predictive classification performance and UQ calibration of the Bayesian baseline while accelerating inference by orders of magnitude.