Infinitesimal Causality

Abstract

Interventions can be varied continuously in many causal models. Differentiating a specified smooth intervention protocol produces vector fields on a statistical model, and their Lie brackets describe the noncommutativity of the corresponding local perturbations. We formulate this differential geometry of interventions on smooth statistical models and call the resulting framework infinitesimal causality (IC). Given a constant-rank distribution spanned by visible intervention fields, we define the normal Lie-bracket residual and show that its vanishing is exactly the involutivity condition in the classical Frobenius theorem. We establish the coordinate invariance of the zero-residual property and characterize its dependence on the intervention protocol, visible span, and metric. Fully observed and latent-variable examples delineate the additional structural assumptions needed to interpret bracket residuals causally. We also distinguish tangent vectors on a statistical parameter manifold from derivatives of stochastic kernels. In the finite-state linearization of a Markov category, normalization and copy compatibility yield well-typed first-order defects. Normalization is automatic for differentiable paths of stochastic kernels, whereas copy compatibility characterizes a more restrictive deterministic or comonoid-preserving perturbation. Together, the geometric and kernel-level constructions make IC a precise foundation for Lie-bracket-based causal diagnostics and identify the assumptions required to pass from local intervention geometry to causal conclusions.

Revised version with substantial corrections and expanded exposition. We clarify the categorical and tangent semantics of ${\mathsf{Stat}}_{\infty }$, and recast the three IC rules as tangent-level compatibility tests. We also add corrected examples, counterexamples delimiting the causal interpretation of Lie-bracket residuals, and new diagrammatic explanations