# Relationship between Causal Calculus (in the sense of the Book of Why) and other existing modeling formalisms?

I am watching this video on youtube: https://youtu.be/zvrcyqcN9Wo?t=2896

about Causal Calculus (CC), namely this section on causal graphs, and it seems to me that this theory of causality is not that different from dynamical systems theory, specifically stochastic differential equations (SDE), except maybe for the fact that dif. eq. encode the causal relationships as derivatives where these are functions of the system state variables whereas in that video the relationships are direct and not implicitly modelled via derivatives... What are the differences between CC and SDE? And what about Markov chains?

It seems to me this is just dynamical systems theory with noise added to the mix... yet given the hype around the topic of Causal Calculus I am wondering if I am overlooking something about CC.

Your question is a bit hard to answer, because it does not make sense to compare causal calculus (a set of inference rules to manipulate causal models) to differential equations (a type of model). So first let's make clear what the causal calculus (or the do-calculus) is.

The do-calculus is a set of three inference rules that help manipulating probabilistic sentences involving interventions in structural causal models. The three rules determine:

• when you can insert or delete observations in probability statements (rule 1);

• when you can exchange action for observation in probability statements (rule 2); and,

• when you can insert or delete actions in probability statements (rule 3).

The conditions for applications of these rules depend on the assumptions of your causal model, in this case encoded by the DAG. The goal of having these rules is to "massage" the query, which is usually an interventional expression, into a purely observational expression (which can then be computed from observational data). The problem of deciding whether your causal assumptions are enough to express the causal query only in terms of available data is what we call the identification problem.

Thus, it doesn't make a lot of sense to ask what is the difference of causal calculus and differential equations, since they are not even in the same class of objects. What might make sense, though, is to ask what is the difference between structural causal models and differential equations. And here I think we can make two distinctions regarding: (i) the level of representation of the phenomenon; and, (ii) what types of problems each literature is trying to solve, and what type of tools they have developed.

The short answer for the first question is that, usually, the models are describing the phenomenon at different levels of abstraction (DE describe continuous changes, whereas SCM equilibrium states). The literature bridging both representations is fairly recent, you may check here, here and here for more discussion.

Regarding the second question, note the causal modeling literature usually is trying to answer questions of the type, "what can we learn from a nonparametric partial specification of a causal model (encoded by a DAG, for instance) and the data? Is that sufficient to answer your causal query?" This is a different problem than, say, assuming to have a known system of differential equations for which you only need to estimate some parameters from data, but for which there's no identification problem involved.

• The last sentence is wrong or at least ambiguous: identification is a well-known problem for the estimation of the parameters of an ODE system. I guess the problem is that causal inference stole a term with a well-defined meaning in statistical inference, and used it to mean something else (I never completely understood the difference). You should clarify that there is an identifiability problem for the estimaction of DEs, but it means something else (I guess?). Also, it's not true that estimating DEs always means to just estimate parameters of a system with a known structure: there is a 1/ Mar 17, 2019 at 7:48
• 2/ growing literature on estimating the structure of DE systems ("dynamical systems") from data. See, e.g., arxiv.org/abs/1902.11136 Mar 17, 2019 at 7:49
• @DeltaIV You’re right, it can be confusing, I will rephrase it later. Mar 17, 2019 at 15:35
• @DeltaIV regarding the text you linked it is not doing anything structural however. Mar 17, 2019 at 15:44
• @DeltaIV what would be the best reference for identification in ODE systems? Mar 18, 2019 at 14:39