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In Pearl "Causality: Models...", he defines Causal Structure in (2.2.1) in terms of "variables" and "functional relationships". This language conflicts with standard mathematical language where a functional is a map from either a vector space or a function to a field like $\mathbb{R}$. I can guess that a random variable is a map from an event space to $\mathbb{R}$.

Can someone please explain what he means in terms of sets and functions?

user442920
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  • The functional relationship is like a predictive model but for the counterfactual. Rubin's term for this was potential outcomes. Mathematically it's like a conditional mean model, except that the causal variables themselves are a network. Does that answer your question? What is the conflict that you perceive? – AdamO Nov 16 '18 at 15:39
  • @AdamO Could this be broken down into sets and functions? That is the basic language of mathematics. – user442920 Nov 16 '18 at 15:41
  • The RHS would be a probability model for the potential outcome--which is a set and a function. The LHS would be a possibly complex non-linear combination of covariates. Read about some models like GAMs for a sense of just how complex they can be. The idea behind Pearl's writing is (quoting him) "to exonerate [himself] from the actual specification and identification of such models". – AdamO Nov 16 '18 at 15:53

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You are conflating the adjective functional with the noun functional. In this particular quote, functional relationship simply means an arbitrary function.

As you said, a functional (substantive) is a special name for a particular type of mapping, for instance, when the domain is a space of functions and the range is the real line. Functionals show up a lot in causal inference as well. Estimands, like $\sum_{z}P(y|x, z)P(z)$, for instance, are functionals --- they assign a real number to any observed joint probability distribution $P(y, x, z)$ (a function).

You can see this use of functionals here which also shows up in Causality, p. 193.

Carlos Cinelli
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