Question: What is the set C in definition 6? Is it C = V \ (X \cup Y)? Or is the statement supposed to hold for all or for one C \subset V \ (X \cup Y)?
Evidence: Sufficient Explanations generalize the previous point by offering settings X = x that causally suffice for Y = y. Existing approaches come in two flavors. Some approaches define sufficient explanations using Pearl's notion of "probability of sufficiency", the deterministic version of which can be informally stated as: if we set the variables in X to x and do not intervene on any of the other variables, then Y takes on the values y. 3 Elsewhere I have called this interpretation of sufficiency weak sufficiency.
Other recent approaches take inspiration from logic and define sufficient explanations of an output as its prime implicants, giving so-called PI-explanations. Informally: if we set the variables in X to x, then Y takes on the values y, regardless of the values of all other variables. In the context of causal models I have coined this direct sufficiency.
Obviously the second form of sufficient explanations is stronger. Coming back to our example, X 2 = 45, 001 is a sufficient explanation for Y = 1 according to both readings. But (X 1 = 50, 000, X 3 = 25, 000) is a sufficient explanation of Y = 1 only on the first reading, for although it is weakly sufficient for Y = 1 it is not directly sufficient. Which one of these notions is correct here? Imagine that a prospective applicant with these values is told that their income and initial deposit suffice for getting a loan. As a result the applicant concludes that there is no need to have such high savings and decides to spend 20, 000, so that their loan application is denied. The applicant would be quite right to be upset about this! Such misunderstandings cannot occur when using direct sufficiency, as that gives us settings whose explanatory value is immune to the influence of interventions on other variables. Instead of concluding from this that one should always rely on direct sufficiency, I propose a generalization of sufficient explanations that adds an element to inform us explicitly as to which variables are assumed to be safeguarded from interventions. Concretely, in addition to specifying which variables need to be set to particular values, a sufficient explanation should also specify a set of variables N that are not to be manipulated for the explanation to be action-guiding. Informally, if we set the variables in X to x and the variables in N are safeguarded from interventions, then Y takes on the value y, regardless of the values of all remaining variables. I call the relevant notion of causal sufficiency at work strong sufficiency, which can be formally defined as follows:
Note that in this definition N cannot just be any set, but rather we require that it is itself entirely determined by X = x. This is because the variables in N can be thought of as a network that transmits the causal influence of X to Y , and the idea of safeguarding this network is that it can continue fulfilling this role even when intervening on X. (I refer the reader to for an elaborate discussion of this definition as well as an equivalent alternative formulation.)
The following straightforward result shows the relative strengths of the above three notions of sufficiency.
Proposition 8 If X = x is directly sufficient for Y = y then X = x is strongly sufficient for Y = y along some N , and if X = x is strongly sufficient for Y = y along some N then X = x is weakly sufficient for Y = y.
Using strong sufficiency we can define a sufficient explanation so that it specifies all the required elements for it to be action-guiding.
If N = Y we speak of a direct sufficient explanation.
Of course we do not want to add redundant parts to a sufficient set, as a good explanation should be as concise as possible. Therefore a good sufficient explanation ought to be minimal with respect both to X and to N .
This allows us to define what makes for a good sufficient explanation of an observed output.
Definition 11 An actual sufficient explanation (X 1 = x 1 , N 1 ) of Y = y in (M, u) is good if it is not dominated by any other actual sufficient explanation in (M, u).
To conclude the analysis of sufficient explanations, I offer a first result that shows the extreme limitation Independence poses. We have seen three very different notions of sufficiency, and yet under Independence they all collapse into one.
Theorem 12 If a causal model M that agrees with h satisfies Independence then the following statements are all equivalent:
(Proofs of all Theorems are to be found in the Supplementary Material.)
Definition 6 X = x is directly sufficient for Y = y in M if for all c ∈ R(C), where C = V − (X ∪ Y ), and all u ∈ R(U) we have that (M,u) |= [X ← x, C ← c]Y = y.
Answer: That was a bit sloppy… The set C is implicitly defined when mentioning c as belonging to the range of V - (X u Y). I will make this explicit.