# Differential Equations vs. Structural Causal Models.

St John Sep 10, 2022 · 2 mins read

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 $\dot{X}_i(t) = f_i(\boldsymbol{X}_{pa_{\mathcal{D}}(i)}), \quad X_i(0) = (\boldsymbol{X}_0)_i \quad \forall i \in \mathcal{I}.$