Intervening is Conditioning

The thesis of this short note is that post-intervention probabilities can be considered to be certain conditional probabilities. Acyclic causal modelsLet us consider an acyclic causal model M of the sort that is central to causal modeling (Spirtes et al. 1993/2000, Pearl 2000/2009, Halpern 2016, Hitchcock 2018).Readers familiar with them can skip this section.M = ⟨S, F ⟩ is a causal model if, and only if, S is a signature and F = {F 1 , . . ., F n } represents a set of n structural equations, for a finite natural number n. S = ⟨U, V, R⟩ is a signature if, and only if, U is a finite set of exogenous variables, V = {V 1 , . . ., V n } is a set of n endogenous variables that is disjoint from U, and R : U ∪ V → R assigns to each exogenous or endogenous variable X in U ∪ V its range (not co-domain) R (X) ⊆ R. F = {F 1 , . . ., F n } represents a set of n structural equations if, and only if, for each natural number i, 1 ≤ i ≤ n: F i is a function from the Cartesian product W i = × X∈U∪V\{V i } R (X) of the ranges of all exogenous and endogenous variables other than V i into the range R (V i ) of the endogenous variable V i .The set of possible worlds of the causal model M is defined as the Cartesian product W = × X∈U∪V R (X) of the ranges of all exogenous and endogenous variables.A causal model M is acyclic if, and only if, it is not the case that there are m endogenous variables V i1 , . . ., V im in V, for some natural number m, 2 ≤ m ≤ n, such that the value of F i(j+1) depends on R V i j for j = 1, . . ., m -1, and the value of F i1 depends on R (V im ).Importantly, dependence is just ordinary functional dependence: F i depends on R V j if, and only if, there are arguments ⃗ w i and ⃗ w i ′ in the domain W i = × X∈U∪V\{V i } R (X) of F i that differ only in the value from R V j such that their values under F i differ, F i ⃗ w i F i ⃗ w i ′ .

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