  We study efficient estimation of an interventional mean associated with a
point exposure treatment under a causal graphical model represented by a
directed acyclic graph without hidden variables. Under such a model, it may
happen that a subset of the variables are uninformative in that failure to
measure them neither precludes identification of the interventional mean nor
changes the semiparametric variance bound for regular estimators of it. We
develop a set of graphical criteria that are sound and complete for eliminating
all the uninformative variables so that the cost of measuring them can be saved
without sacrificing estimation efficiency, which could be useful when designing
a planned observational or randomized study. Further, we construct a reduced
directed acyclic graph on the set of informative variables only. We show that
the interventional mean is identified from the marginal law by the g-formula
(Robins, 1986) associated with the reduced graph, and the semiparametric
variance bounds for estimating the interventional mean under the original and
the reduced graphical model agree. This g-formula is an irreducible, efficient
identifying formula in the sense that the nonparametric estimator of the
formula, under regularity conditions, is asymptotically efficient under the
original causal graphical model, and no formula with such property exists that
only depends on a strict subset of the variables.
