[Warning: “cyclic” overload. I think in this post it’s referring to the dynamical systems definition, i.e. variables reattain the same state later in time. I’m referring to Pearl’s causality definition: variable X is functionally dependent on variable Y, which is itself functionally dependent on variable X.]
Turns out Chaos is not Linear...
I think the bigger point (which is unaddressed here) is that chaos can’t arise for acyclic causal models (SCMs). Chaos can only arise when there is feedback between the variables right? Hence the characterization of chaos is that orbits of all periods are present in the system: you can’t have an orbit at all without functional feedback. The linear approximations post is working on an acyclic Bayes net.
I believe this sort of phenomenon [ chaos ] plays a central role in abstraction in practice: the “natural abstraction” is a summary of exactly the information which isn’t wiped out. So, my methods definitely needed to handle chaos.
Not all useful systems in the world are chaotic. And the Telephone Theorem doesn’t rely on chaos as the mechanism for information loss. So it seems too strong to say “my methods definitely need to handle chaos”. Surely there are useful footholds in between the extremes of “acyclic + linear” to “cyclic + chaos”: for instance, “cyclic + linear”.
At any rate, Foundations of Structural Causal Models with Cycles and Latent Variables could provide a good starting point for cyclic causal models (also called structural equation models). There are other formalisms as well but I’m preferential towards this because of how closely it matches Pearl.
As I understand it, the proof in the appendix only assumes we’re working with Bayes nets (so just factorizations of probability distributions). That is, no assumption is made that the graphs are causal in nature (they’re not necessarily assumed to be the causal diagrams of SCMs) although of course the arguments still port over if we make that stronger assumption.
Is that correct?