Causal Discovery with False Positive Error Control

Causal discovery methods produce data-driven hypotheses about causal relationships, which may become new scientific findings or inform decisions about downstream interventions. In such settings, false positive causal claims can be more consequential than false negatives. Yet few methods target false positive error control in causal discovery, and existing error guarantees are often asymptotic and require strong versions of the faithfulness assumption. We introduce the $m$-conservative PC algorithm and prove that it controls false discovery rate in the linear Gaussian setting under an extremely weak version of the faithfulness assumption. In our framework, false positive error for causal graphs is defined based on the conditional-independence information they entail, yielding a metric that penalizes both false adjacencies and incorrect edge orientations. The parameter $m$ quantifies a trade-off between algorithmic complexity and the strength of the required faithfulness assumption. The $m$-conservative PC algorithm combines undirected graph learning, a Benjamini-Yekutieli-style procedure for conditional independence testing, and a conservative orientation step that marks unresolved or contradictory edge directions as ambiguous. Our algorithm produces valid outputs and retains error guarantees even in the finite-sample setting, despite possible random errors in the conditional independence tests.