Necessary and sufficient graphical conditions for optimal adjustment sets in causal graphical models with hidden variables

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