Causal models for dynamical systems

TL;DR

This paper extends structural causal models to continuous-time dynamical systems, introducing causal kinetic models that incorporate interventions directly into differential equations for better modeling of system evolution.

Contribution

It proposes a novel extension of causal models to dynamical systems, focusing on continuous-time models and defining interventions within differential equations.

Findings

Introduces two types of causal kinetic models with different noise assumptions

Defines interventions in the differential equations of dynamical systems

Focuses on modeling the full time evolution rather than stationary distributions

Abstract

A probabilistic model describes a system in its observational state. In many situations, however, we are interested in the system's response under interventions. The class of structural causal models provides a language that allows us to model the behaviour under interventions. It can been taken as a starting point to answer a plethora of causal questions, including the identification of causal effects or causal structure learning. In this chapter, we provide a natural and straight-forward extension of this concept to dynamical systems, focusing on continuous time models. In particular, we introduce two types of causal kinetic models that differ in how the randomness enters into the model: it may either be considered as observational noise or as systematic driving noise. In both cases, we define interventions and therefore provide a possible starting point for causal inference. In this sense, the book chapter provides more questions than answers. The focus of the proposed causal kinetic models lies on the dynamics themselves rather than corresponding stationary distributions, for example. We believe that this is beneficial when the aim is to model the full time evolution of the system and data are measured at different time points. Under this focus, it is natural to consider interventions in the differential equations themselves.