Causal Density Functions

Abstract

We study the full density ratio between a specified intervention regime $P_a$ and an observational regime $P_0$, $\ddot{I}{}_a=d{P}_a/d{P}_0$, under the prerequisite $P_a\ll P_0$. We call the regime-indexed ratio a causal density function when $P_a$ is an identified or directly observed interventional law. The underlying Radon-Nikodym derivative and the identity ${\mathbb{E}}_a[f(Z)]={\mathbb{E}}_0\left[f(Z)\ddot{I}{}_a(Z)\right]$ are classical importance weighting, not new identification results. Our narrower question is whether retaining the entire pointwise ratio is useful as a reusable diagnostic across several downstream functionals. We evaluate a two-density plug-in baseline through held-out moment transport and overlap stress tests on synthetic and perturbation data. We also report a pairwise graph-scoring heuristic as a negative result: its F1 is \(0.10\) on a synthetic DAG, \(0.12\) on Sachs, and \(0.33\) on a multi-regime chain. These experiments do not establish an estimation advantage over direct density-ratio, inverse-probability, Riesz, or doubly robust methods; they instead delimit what the pointwise ratio target and the present plug-in estimator do and do not provide.

Substantially revised and narrowed the paper: removed the erroneous categorical theorems; corrected the intervention, domination, convergence, and density-ratio claims; repositioned the estimator as a baseline; corrected reported experimental values and citations; and documented the pairwise graph scorer as a negative result. A detailed errata/change log appears in the supplement