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