Causal Density Functions

2026-05-30

AI Summary

This work addresses the limitation of existing causal strength measures, which primarily focus on global distributional shifts under intervention and fail to capture local causal effects. The authors propose a causal density function defined as the Radon–Nikodym derivative between interventional and observational distributions, introducing for the first time a pointwise causal density ratio that models causal effects as a local measure transformation. They establish a reweighting identity linking this ratio to do-expectations and, leveraging a formal framework of conditioning and intervention grounded in Kan semantics, construct an estimable causal density function along with a do-curve estimator. Experiments on both synthetic and real perturbation data demonstrate that the method effectively reproduces interventional expectations through reweighting of observational data, enabling fine-grained and verifiable causal inference.

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.

Problem

• causal density functions
• interventional laws
• observational laws
• density ratios
• causal effects

Innovation

• causal density functions
• Radon-Nikodym derivative
• interventional distributions
• do-calculus
• density ratio estimation