Acceptance-Side Multiple Testing for Proxy Discovery

Proximal causal learning (PCL) identifies causal effects under unmeasured confounding from valid proxies of the confounder. When proxies must be discovered from high-dimensional data, thousands to millions of candidate proxy sets are screened at once, and the consequential error is false validation: retaining an invalid candidate that then biases the causal estimate. We formulate proxy discovery as an acceptance-side multiple-testing problem. A joint tetrad test gives one $p$-value per candidate triplet, a two-group model converts these into posterior validity scores $L(p)=\mathbb{P}(H=0\mid P=p)$, and accepting $L(p)\ge \tau$ has oracle false validation rate (FVR) at most $1-\tau$. The plug-in rule inherits no guarantee: where invalid $p$-values retain mass near one, the upward bias of Storey's ${\hat{\pi }}_0$, conservative for the FDR, is anti-conservative for the FVR, and no rule based on the $p$-values alone can correct it. An exact identity equates the accepted set's mean absolute bias to its empirical FVR times the mean bias of the accepted invalid triplets. Simulations show where each layer holds and fails; we illustrate the pipeline on 5.7 million candidate triplets from TCGA–BRCA.