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We study the selection of adjustment sets for estimating the interventionalmean under an individualized treatment rule. We assume a non-parametric causalgraphical model with, possibly, hidden variables and at least one adjustmentset comprised of observable variables. Moreover, we assume that observablevariables have positive costs associated with them. We define the cost of anobservable adjustment set as the sum of the costs of the variables thatcomprise it. We show that in this setting there exist adjustment sets that areminimum cost optimal, in the sense that they yield non-parametric estimators ofthe interventional mean with the smallest asymptotic variance among those thatcontrol for observable adjustment sets that have minimum cost. Our results arebased on the construction of a special flow network associated with theoriginal causal graph. We show that a minimum cost optimal adjustment set canbe found by computing a maximum flow on the network, and then finding the setof vertices that are reachable from the source by augmenting paths. Theoptimaladj Python package implements the algorithms introduced in this paper.

A note on efficient minimum cost adjustment sets in causal graphical models