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.

翻译：我们引入因果密度函数：一种比较干预分布与观测分布的Radon-Nikodym导数，从而作为因果效应的局部密度比。不同于众多因果强度度量对图形干预后的整体分布进行比较，因果密度函数提供了一种逐点测度变换对象，可被估计、校准并用于定向影响评分。基本恒等式 \[ \mathbb{E}_{\mathrm{do}}[f(Y)] = \mathbb{E}_{\mathrm{obs}}\!\left[f(Y)ρ(X,Y)\right] \] 使因果密度直接可检验：若估计的密度比正确，经$ρ$重新加权的观测期望可再现干预期望。我们推导出面向do曲线和定向边评分的实用估计量，将这一构造与条件作用和干预的Radon-Nikodym/Kan语义相关联，并在合成和真实扰动基准数据集上评估所提估计量。