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 is well matched to 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.
Highlights
We extend the causal emergence framework to continuous systems and show that in that context, it is naturally related to information geometry and sloppy models
The matrix product g−1 h appearing in Equation (8) is a linear transformation, and its eigenvalues are coordinate independent. This way, to evaluate how sloppy a given causal model is, we suggest that it is more appropriate to study the eigenvalues of h−1 g instead of those of g as is usually done [6]
Solid objects are made of many loosely-connected atoms, yet in our everyday experience, we invariably view them as single units
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. We extend the causal emergence framework to continuous systems and show that in that context, it is naturally related to information geometry and sloppy models This leads to a novel construction, which we term causal geometry, where finding the causally most informative model translates to a geometric matching between our intervention capabilities and the effects on system behaviors, both expressed as distance metrics on the model’s parameter space. This framework captures precisely how the inherent properties of the system’s behavior (its sloppiness structure) and their relation to its use context (matching to intervention capabilities) both play a role in optimal model selection, thereby reconciling the two above perspectives.
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