Total: 1
Causal effects estimated by covariate adjustment on a discovered graph have invalid confidence intervals, because the graph-selection step is ignored. We develop a unified selective-inference framework. For constraint-based discovery with latent confounders (FCI) under Gaussianity, we prove that the selection event, given the execution trace and signs, is a polyhedron in Fisher-z space; along the inference direction it becomes polynomial/rational constraints solving to a union of intervals. Inverting the truncated-Gaussian approximate pivot gives exact finite-sample $1-\alpha$ coverage, with known $\sigma^2$, of the adjustment functional $\gamma_1(S^\star)$ when $\{i\}\cup S^\star$ contains the outcome's Markov blanket, and approximate coverage otherwise, with a non-vanishing distortion governed by the variance-ratio excess $\rho^2=\sigma^2_{j|X}/\sigma^2_{j|-j}-1$ (small under sparsity); the $\hat\sigma^2$ plug-in does not remove it. This is the first truncation-set characterization handling latent confounders. Coverage of the structural effect $\beta_{i\to j}$ is asymptotically valid up to the same distortion when the adjustment set is valid (exact when $\rho^2=0$), via high-dimensional FCI consistency under $d=o(\sqrt n)$. For unmodified GES, heuristic Taylor linearization indicates approximate coverage $1-\alpha-O(d^2/\sqrt n)$ when $n\gg d^4(\log d)^2$. In the nonparametric setting, $\Theta(n^{3/4})$-split discovery incurs width ratio $1+\Theta(n^{-1/4})$ versus a known-graph oracle.