Causal Geometry

TL;DR

Causal geometry extends information geometry to include causal interventions, providing a geometric measure of how well models match available interventions, and revealing that coarse-grained models can sometimes be more informative than microscopic ones.

Contribution

This paper introduces causal geometry, a novel framework integrating causation into information geometry, and defines a geometric measure of effective information based on intervention matching.

Findings

Effective information is quantified by geometric congruence between effects and interventions.

Causal emergence shows coarse models can be more informative than microscopic models.

Matching interventions with effects enhances model informativeness.

Abstract

Information geometry has offered a way to formally study the efficacy of scientific models by quantifying the impact of model parameters on the predicted effects. However, there has been little formal investigation of causation in this framework, despite causal models being a fundamental part of science and explanation. Here we introduce causal geometry, which formalizes not only how outcomes are impacted by parameters, but also how the parameters of a model can be intervened upon. Therefore we introduce a geometric version of "effective information" -- a known measure of the informativeness of a causal relationship. We show that it is given by the matching between the space of effects and the space of interventions, in the form of their geometric congruence. Therefore, given a fixed intervention capability, an effective causal model is one that matches those interventions. This is a consequence of "causal emergence," wherein macroscopic causal relationships may carry more information than "fundamental" microscopic ones. We thus argue that a coarse-grained model may, paradoxically, be more informative than the microscopic one, especially when it better matches the scale of accessible interventions -- as we illustrate on toy examples.