Causal Density Functions

Abstract

We introduce causal density functions: Radon-Nikodym derivatives that compare interventional laws to observational laws and therefore act as local density ratios for causal effects. Whereas many causal-strength measures compare whole distributions after graph surgery, causal density functions provide a pointwise change-of-measure object that can be estimated, calibrated, and used to score directed influence. The basic identity \[ \mathbb{E}_{\mathrm{do}}[f(Y)] = \mathbb{E}_{\mathrm{obs}}\!\left[f(Y)ρ(X,Y)\right] \] makes causal density directly testable: if the estimated density ratio is correct, observational expectations reweighted by $ρ$ reproduce interventional expectations. We derive practical estimators for do-curves and directed edge scores, relate the construction to Radon-Nikodym/Kan semantics for conditioning and intervention, and evaluate the resulting estimators on synthetic and real perturbation benchmarks.