  The problem of selecting optimal valid backdoor adjustment sets to estimate
total causal effects in graphical models with hidden and conditioned variables
is addressed. Previous work has defined optimality as achieving the smallest
asymptotic variance compared to other adjustment sets and identified a
graphical criterion for an optimal set for the case without hidden variables.
For the case with hidden variables currently a sufficient graphical criterion
and a corresponding construction algorithm exists. Here optimality is
characterized by an information-theoretic approach based on the mutual
informations among cause, effect, adjustment set, and conditioned variables.
This characterization allows to derive the main contributions of this paper: A
necessary and sufficient graphical criterion for the existence of an optimal
adjustment set and an algorithm to construct it. The results are valid for a
class of estimators whose variance admits a certain information-theoretic
decomposition.
