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

The problem of selecting optimal valid 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 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 conditional 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 a definition and algorithm to construct it. Further, the optimal set is valid if and only if a valid adjustment set exists and has smaller (or equal) asymptotic variance compared to the Adjust-set proposed in Perkovic et al. (2018) (arXiv:1606.06903) for any graph, whether graphical optimality holds or not. The results are valid 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. For estimators outside of the class studied here none of the considered adjustment sets outperforms all others, but a minimized variant of the optimal set proposed here tends to have lower variance. Surprisingly, among the randomly created setups more than 80\% 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}.