[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.
Yeah, the chaos piece predated the Telephone Theorem. The Telephone Theorem does apply just fine to chaotic systems (the Bayes Net just happens to have time symmetry), but it’s way more general.
[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.]
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.
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.
Yeah, the chaos piece predated the Telephone Theorem. The Telephone Theorem does apply just fine to chaotic systems (the Bayes Net just happens to have time symmetry), but it’s way more general.