Causal Geometry

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.

Emergence and Causality in Complex Systems: A Survey of Causal Emergence and Related Quantitative Studies

“…Second, knowledge of the system's transition probability matrix (TPM) is required to calculate the effective information metrics. Efforts have been made to apply or extend the concepts of effective information to continuous systems [26,31,100].…”

“…Alternatively, Pavel Chvykov and Erik Hoel extended the causal emergence framework to continuous systems and proposed the concept of causal geometry [26].…”

“…In [26], the authors considered a more general case in which intervention noise is added to the input (intervention) variable x in Equation (19). The intervention noise is denoted as ξ ∼ N (0, δ 2 ), where δ is the standard deviation of ξ.…”

“…As demonstrated in the previous subsection, the measurement of causal emergence by vanilla effective information [19] is defined for stationary dynamics systems in a selfsupervised fashion since the input (cause) space and the output (effect) space share the support. However, Hoel generalized effective information to general input-output systems [26] (see Section 3.5.5), where the cause and the effect space take arbitrary support, which could either be discrete or continuous. Empowered by the general notation of EI, causal emergence could be applied in supervised machine learning to evaluate the strength of causation between the feature space X (or its learned representation) and the target space Y, thus enhancing the prediction from the cause (feature) to the effect (target).…”

“…Several examples of discrete Markov chains have been shown to exhibit causal emergence if their EI values of macrolevel dynamics are larger than those of micro-level dynamics. Hoel and other researchers further extended the measures of effective information and causal emergence for dynamical systems with continuous variables [26] and complex networks [27]. Other measures are also possible for quantifying causal effects and, consequently, causal emergence.…”