  In order to achieve unbiased and efficient estimators of causal effects from
observational data, covariate selection for confounding adjustment becomes an
important task in causal inference. Despite recent advancements in graphical
criterion for constructing valid and efficient adjustment sets, these methods
often rely on assumptions that may not hold in practice. We examine the
properties of existing graph-free covariate selection methods with respect to
both validity and efficiency, highlighting the potential dangers of producing
invalid adjustment sets when hidden variables are present. To address this
issue, we propose a novel graph-free method, referred to as CMIO, adapted from
Mixed Integer Optimization (MIO) with a set of causal constraints. Our results
demonstrate that CMIO outperforms existing state-of-the-art methods and
provides theoretically sound outputs. Furthermore, we present a revised version
of CMIO capable of handling the scenario in the absence of causal sufficiency
and graphical information, offering efficient and valid covariate adjustments
for causal inference.

Methodology