Title: Causal Density Functions

Abstract:

This work presents causal density functions, defined as Radon-Nikodym derivatives that serve to contrast interventional distributions with observational ones. These functions operate as local density ratios for causal effects. Unlike conventional metrics for causal strength, which typically evaluate entire distributions following graph modification, causal density functions offer a pointwise change-of-measure framework. This approach facilitates estimation, calibration, and the scoring of directed influence. The fundamental identity $\mathbb{E}{\mathrm{do}}[f(Y)] = \mathbb{E}{\mathrm{obs}}!\left[f(Y)\rho(X,Y)\right]$ renders causal density empirically testable; specifically, if the estimated density ratio is accurate, observational expectations reweighted by $\rho$ will yield interventional expectations. We develop practical estimators for do-curves and directed edge scores, connect this construction to Radon-Nikodym and Kan semantics regarding conditioning and intervention, and assess the performance of these estimators using both synthetic data and real-world perturbation benchmarks.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC