Forest-structured Dvoretzky-Kiefer-Wolfowitz post hoc confidence bounds

⚠️ Curated Limitations & Negative Results

• Exact equality V^*_{R_ref}(S) = V_tilde_{R_ref}(S) = V_tilde^d_{R_ref}(S) and the O(Hm) bottom-up dynamic program (Theorem 3.6, Algorithm 1) require the reference family R_ref to satisfy the (Forest) condition (any two regions are disjoint or nested); when reference regions overlap without nesting, V^*_{R_ref}(S) < V_tilde_{R_ref}(S) can be strict (Example 2.2, page 4, verified in research/discovery/forest-structured-dkw-post-hoc-confidence-bounds-boundaries.py where V^*_{R_ref}(S) = 1 < V_tilde_{R_ref}(S) = 2).
• The local DKW-Massart calibration (Proposition 4.1) requires within-region independence of null p-values (Indep) and incurs an unavoidable non-zero floor cost zeta_k(X) >= floor(log(K/alpha)/2) >= 1 whenever alpha <= e^{-2}*K (page 9), making pure DKW bounds trivial on small selection sets S of size |S| <= floor(log(K/alpha)/2) unless combined with Simes via the hybrid bound V^gamma_{hybrid} (Section 5.3, pages 12-14).
• Four minor typographical slips in arXiv:1807.01470v2 were identified and verified during full-text inspection: (1) in Section 2.2 (page 4, line 1), 'for all S subset N_m, V^*_{R_ref}(R) <= V(R)' writes R instead of S inside V^*_{R_ref} and V, (2) in Equation (16) and Remark 4.2 (page 9), the integer floor brackets floor(...)^2 are printed inside the exponent 2 rather than enclosing the squared expression floor((...)^2) derived in Equation (32) of the proof of Proposition 4.1 (page 18; note also page 9 line 13 writes zeta_k(X) >= floor(log(K/alpha)/2) = floor(C^2), which is the floor of the square rather than the square of the floor), (3) in Section 5.2 (page 12, line 1), 'choice of the model parameters d (in rows)' writes d instead of r (the signal proportion parameter shown in the row labels of Figure 8), and (4) in Section 6.2 Equation (24) (page 15), P_{X~P}(|R_k cap H_0(P)| <= L_k(lambda)) <= lambda writes '<= L_k(lambda)' inside the probability instead of '> L_k(lambda)' (or '>= 1 - lambda' on the right-hand side).
• All 24 pages of the full-text PDF of Durand, Blanchard, Neuvial, & Roquain (2020, Scandinavian Journal of Statistics 47(4):1114-1148, arXiv:1807.01470v2, DOI:10.1111/sjos.12453) were read. Constructed numerical checks in research/discovery/forest-structured-dkw-post-hoc-confidence-bounds-boundaries.py verify (1) Example 2.2 non-forest gap V^* = 1 < V_tilde = 2, (2) Example 3.2/3.4 node depths, Algorithm 2 atomic partition, and Theorem 3.6 / Algorithm 1 exactness across all 256 subsets of an 8-element forest, (3) Proposition 4.1 empirical JER control and Remark 4.3 / Lemma A.1 strict dominance over the Genovese-Wasserman (2004) DKW bound across 3,000 region-trial checks, and (4) Proposition 4.5 / Figure 6 localized-signal ratio upper bounds and Section 5.3 Simes-DKW hybrid envelope complementarity.