Robust Confounder Selection Under Imperfect Primary-Set Elicitation

Fix two distinct observed variables $X$ (treatment) and $Y$ (outcome). Let $\tilde G$ be a causal directed acyclic graph on vertex set $V\cup U$, where $V$ are observed variables (including $X,Y$) and $U$ are latent (unobserved) variables. For any observational distribution $P$ that is Markov and faithful to $\tilde G$ and satisfies positivity, call a set $S\subseteq V\setminus\{X,Y\}$ a valid covariate-adjustment set for the total causal effect of $X$ on $Y$ if the adjustment formula

holds for all $x,y$ (with integral form for continuous $S$) for every such $P$. Let

Consider an interactive algorithm that can ask at most $B$ queries. At round $t$, there is a target set $P_t^\star$ (the exact primary set that would be provided in the noiseless/oracle procedure), but the algorithm receives a noisy response $\widehat P_t$. Let $H_{t-1}$ be the full interaction history before round $t$. Two noise models of interest are: (i) mis-elicitation probability bound $\Pr(\widehat P_t\neq P_t^\star\mid H_{t-1})\le \varepsilon$ for all $t$; (ii) bounded set-error model with metric $d(A,B):=|A\triangle B|$ and either deterministic bound $d(\widehat P_t,P_t^\star)\le \delta$ or high-probability bound $\Pr(d(\widehat P_t,P_t^\star)>\delta\mid H_{t-1})\le \varepsilon$.