Causal Density Functions

We study the full density ratio between a specified intervention regime Pa and an observational regime P0, ρa=dPa/dP0, under the prerequisite Pall P0. We call the regime-indexed ratio a causal density function when Pa is an identified or directly observed interventional law. The underlying Radon-Nikodym derivative and the identity mathbbEa[f(Z)] = mathbbE0!left[f(Z)ρa(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.