提出因果密度函数概念，用于诊断干预效果的可迁移性。
Causal Density Functions
- 定义因果密度函数为干预与观测分布的点对点密度比。
- 实验显示该函数在合成数据和扰动数据上表现有限。
- 适合关注因果推断中干预诊断的研究者参考。
我们研究了在 $P_a\ll P_0$ 前提下，特定干预分布 $P_a$ 与观测分布 $P_0$ 之间的完整密度比 $ρ_a = dP_a/dP_0$。当 $P_a$ 为可识别或直接观测的干预律时，该比率称为因果密度函数。其背后的 Radon-Nikodym 导数及期望等式 $ \mathbb{E}_{a}[f(Z)] = \mathbb{E}_{0}\left[f(Z)ρ_a(Z)\right] $ 属于经典重要性加权，并非新识别结果。本文核心问题在于：保留整个点对点比率是否能作为可复用的诊断工具，适用于多个下游函数。我们在合成数据和扰动数据上通过留出矩传输与重叠应力测试评估了两密度插值基线。此外报告了一种成对图评分启发式方法的负面结果：在合成 DAG 上 F1 为 0.10，在 Sachs 数据集上为 0.12，在多干预链上为 0.33。这些实验未证明该点对点比率目标或当前插值估计器在密度比、逆概率加权、Riesz 或双重稳健方法上具有估计优势；相反，它们界定了该目标与估计器的能力边界。
原文摘要 · Abstract (English)
We study the full density ratio between a specified intervention regime $P_a$ and an observational regime $P_0$, $ρ_a=dP_a/dP_0$, 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)ρ_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.
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