Distributional Sensitivity Analysis: Enabling Differentiability in Sample-Based Inference

Abstract

This work introduces a mathematical framework for estimating the space-parameter sensitivity of random samples in arbitrary dimensions. Such sensitivity effectively acts as gradients of random samples with respect to distributional parameters, which are essential in sample-based inverse problems in nuclear physics, such as inferring quantum correlation functions. We present two analytical formulae for sensitivity and gradient estimation. The first interprets sensitivity as the partial derivatives of the inverse mapping of 1-D conditional distributions. The second, suited for optimization methods that tolerate inexact gradients, applies a diagonal approximation that reduces computational cost with minimal accuracy loss. When closed forms are unavailable, four second-order numerical algorithms are provided to approximate both expressions. Verification and validation studies confirm the correctness of these algorithms and the effectiveness of the proposed formulae. A nuclear physics application demonstrates how the framework enables uncertainty quantification and parameter inference for quantum correlation functions. Unlike existing approaches, our method requires neither model fitting nor knowledge of sampling algorithms or high-dimensional integrals, making it suitable for black-box or simulation-based samplers. Moreover, it renders arbitrary sampling subroutines differentiable, facilitating integration into deep learning and automatic differentiation frameworks. Algorithmic details and open-source implementations are provided to ensure reproducibility and promote further development.