  The problem of selecting optimal backdoor adjustment sets to estimate causal
effects in graphical models with hidden and conditioned variables is addressed.
Previous work has defined optimality as achieving the smallest asymptotic
estimation variance and derived an optimal set for the case without hidden
variables. For the case with hidden variables there can be settings where no
optimal set exists and currently only a sufficient graphical optimality
criterion of limited applicability has been derived. In the present work
optimality is characterized as maximizing a certain adjustment information
which allows to derive a necessary and sufficient graphical criterion for the
existence of an optimal adjustment set and a definition and algorithm to
construct it. Further, the optimal set is valid if and only if a valid
adjustment set exists and has higher (or equal) adjustment information than the
Adjust-set proposed in Perkovi{\'c} et al. [Journal of Machine Learning
Research, 18: 1--62, 2018] for any graph. The results translate to minimal
asymptotic estimation variance for a class of estimators whose asymptotic
variance follows a certain information-theoretic relation. Numerical
experiments indicate that the asymptotic results also hold for relatively small
sample sizes and that the optimal adjustment set or minimized variants thereof
often yield better variance also beyond that estimator class. Surprisingly,
among the randomly created setups more than 90\% fulfill the optimality
conditions indicating that also in many real-world scenarios graphical
optimality may hold. Code is available as part of the python package
\url{https://github.com/jakobrunge/tigramite}.
