发表机构
University of Calgary（卡尔加里大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究因果推断中仅由联合密度和得分场构建的几何无法识别因果方向的问题，开发干预类似物，定义因果影响等，引入干预响应场和因果度量，为相关设计产生几何词典，组织相关关系但未增加基本假设外的识别能力。
AI 中文摘要
设\(p(x)\)为变量\(X\)的联合密度，\(\psi(x)=\nabla_x\log p(x)\)为其得分场。仅由\(p\)和\(\psi\)构建的几何无法识别因果方向，因为具有相同观测分布的结构模型具有相同的得分几何。本文开发了一种干预类似物。硬干预\(\operatorname{do}(X_k=\xi)\)不仅重新加权联合律，还将分布限制在子流形\({x_k=\xi}\)上。因此其得分应在其余\(d - 1\)个自由坐标上定义。定义因果影响\(X_k\rightsquigarrow X_j\)为\(X_j\)的干预边际分布随\(\xi\)的变化，并表明边际干预得分的相应导数给出了影响的局部充分条件。将观测得分投影到可允许的干预方向上通常无法恢复因果响应。因此引入由结构信息提供的干预响应场。因果度量定义为具有共同目标的一族干预上的费希尔信息度量，避免跨目标的不适定比较。该框架为随机试验、工具变量和条件独立设计产生了一个几何词典，阐明了每种方法能识别和不能识别的内容。一个双变量高斯示例给出了两个具有相同观测得分但不同干预得分导数的模型。该框架组织了设计、干预和得分场之间的关系，但在基本假设之外没有增加识别能力。在Pearl的因果阶梯中，观测得分几何属于关联，干预索引得分场属于干预，单位水平反事实几何留待未来工作。
英文摘要
Let $p(x)$ be the joint density of variables $X$, and let $ψ(x)=\nabla_x\log p(x)$ be its score field. Geometry constructed from $p$ and $ψ$ alone cannot identify causal direction: structural models with the same observational distribution have the same score geometry. I develop an interventional analogue. A hard intervention $\operatorname{do}(X_k=ξ)$ does not merely reweight the joint law; it restricts the distribution to the submanifold ${x_k=ξ}$. Its score should therefore be defined on the remaining $d-1$ free coordinates. I define causal influence $X_k\rightsquigarrow X_j$ as variation of the interventional marginal distribution of $X_j$ with $ξ$, and show that the corresponding derivative of the marginal interventional score gives a local sufficient condition for influence. Projecting the observational score onto admissible intervention directions does not generally recover causal response: two models may share the same observational score and admissible set yet respond differently. I therefore introduce an interventional response field supplied by structural information. A causal metric is defined as the Fisher information metric on a family of interventions with a common target, avoiding ill-posed comparisons across targets. The framework yields a geometric dictionary for randomized trials, instrumental variables, and conditional-independence designs, clarifying what each does and does not identify. A bivariate Gaussian example gives two models with the same observational score but different interventional score derivatives. The framework organizes relations among designs, interventions, and score fields, but adds no identification beyond the underlying assumptions. In Pearl's Ladder of Causation, observational score geometry belongs to association, intervention-indexed score fields to intervention, and unit-level counterfactual geometry is left for future work.
