Differential Equations vs. Structural Causal Models.

causality physics statistics

St John St John Sep 10, 2022 · 2 mins read
Differential Equations vs. Structural Causal Models.

The explicit study of causality in AI fields has officially hit the ‘hype cycle’, at least according to Gartner [1]. There are usually important reasons these fields of study gain momentum like this. In AI, hype is usually driven by performance in real-world tasks - think AlphaGO for (deep) reinforcement learning (RL). For causal AI, this appears to be different. The hype seems to be driven by (1) the understanding of the need for causal understanding in AI systems, and (2) the “preaching” of well respected individuals in the field.

Although the study of causality and causal models is becoming mainstream, there seems to be a disconnect between models that fully capture causal theories - differential equations and physical models. At first glance, these concepts seems starkly different, but it makes sense that the models must be equivalent at a causal level if they are modelling the same phenomena. So what gives? What is the link between these ideas? Does one encompass the other?

This article aims to give some intuition for how causal models abstract differential equations. For the most part, we will focus on structural causal models (SCMs) and ordinary differential equations (ODEs). The first thing to note about the (Pearl) view of causal modelling is that there is an emphasis on acyclicity in the models. This makes sense in the context of reasoning in terms of “paths of cause and effect,” in which parent and child variables are considered. However, this acyclic constraint means that it doesn’t immediately extend to systems where feedback is important.

Modelling Cyclicity

Mooij, Janzing, and Schölkopf (2013) [2] consider exactly this. They argue that, of existing causal frameworks, the SCM formulation is easiest to extend to feedback systems by dropping the acyclicity constraint. They show that, when considering underlying systems of ODEs, an alternative interpretation of the SCM emerges naturally.

In my opinion, the most interesting point of this line of work is that a causal graph may simply serve to formalise how intervening on a variable influences the equilibrium state of other variables in the model.

Ordinary Differential Equations

We start with a dynamical system $\mathcal{D}$ of $D$ coupled first-order ODEs, with initial conditions $X_0 \in \mathcal{R}_\mathcal{I}.$ The system is defined as \(\dot{X}_i(t) = f_i(\boldsymbol{X}_{pa_{\mathcal{D}}(i)}), \quad X_i(0) = (\boldsymbol{X}_0)_i \quad \forall i \in \mathcal{I}.\)

References

  • Header Image
  • [1] What’s New in the 2022 Gartner Hype Cycle for Emerging Technologies. (2022). Retrieved 10 September 2022, from https://www.gartner.com/en/articles/what-s-new-in-the-2022-gartner-hype-cycle-for-emerging-technologies
  • [2] Mooij, J., Janzing, D., & Schölkopf, B. (2013). From Ordinary Differential Equations to Structural Causal Models: the deterministic case. Retrieved 10 September 2022, from https://arxiv.org/abs/1304.7920
St John
Written by St John
Author of the Asking Why Blog - a personal blog and website with everything I find interesting.