Selectively traversed accumulation rules for structured false discovery rate control
Mathematical Methods AIMech:SelectivelyTraversedAccumulationRulesForStructuredFDRControl DRAFT
Aliases: STAR Framework for Interactive Structured Multiple Testing, Selectively Traversed Accumulation Rules (STAR), Interactive Masked Peeling for Structured FDR Control, Accumulation-Masked Interactive Peeling Under Structural Constraints
An interactive multiple-testing methodology (Lei, Ramdas, & Fithian 2021, Biometrika 108(2):253-267, arXiv:1710.02776) that controls the false discovery rate (FDR) in finite samples while enforcing arbitrary structural constraints R_t in K on the discovery set (including rooted tree heredity, directed acyclic graph heredity, spatial convexity or connectedness, and polynomially sized group structures). Given a bounded non-decreasing accumulation function h : [0, 1] -> [0, infinity) satisfying int_0^1 h(p) dp = 1, STAR constructs a paired continuous strictly decreasing reflection s : [0, 1] -> [0, 1] via H(s(p)) = H(p) where H(x) = int_0^x (h(p) - 1) dp, and reveals to the analyst only the masked p-values g(p_i) = min{p_i, s(p_i)} together with side information x_i, the unmasked p-values (p_i)_{i notin R_t} outside the current candidate set R_t, and the aggregate false discovery proportion estimate FDP_hat_t = (h(1) + sum_{i in R_t} h(p_i)) / (1 + |R_t|). Starting from R_0 = [n], the analyst iteratively peels R_{t+1} subsetneq R_t inside K using any F_t-measurable model-assisted or human-in-the-loop rule until tau = inf{t : FDP_hat_t <= alpha}, guaranteeing exact finite-sample conditional FDR control E[FDP(R_tau) | (x_i, g(p_i))_{i=1}^n] <= alpha via optional stopping (Theorem 2.3).
Identity Notes & Concept Scope
No normalized identity match for selectively traversed accumulation rules or interactive structured FDR control (STAR) across all 442 pinned AIO records and 586 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches across data, sources, research, history, and reviews returned only frontier references in the Round 191 discovery report. Distinct from AIMech:AdaptivePValueThresholding (Lei & Fithian 2018, AdaPT), which adapts coordinatewise rejection thresholds s(x_i) to covariates via symmetric mirror reflection min(p_i, 1 - p_i) without coupling general accumulation functions h(p) to reflection maps s(p) via H(s(p)) = H(p) or enforcing hard combinatorial structural constraints R_t in K (trees, DAGs, spatial regions) through boundary peeling and one-group EM working models; distinct from AIMech:FocusedBH (Katsevich, Sabatti, & Bogomolov 2023), which applies an upfront monotonic structural filter F(P, alpha) to unmasked p-values and calibrates a single scalar p-value cutoff via permutation simulation rather than interactive partial-masking peeling adapted to F_t; and distinct from AIMech:SimultaneousPathwiseFalseDiscoveryProportionBounds (Katsevich & Ramdas 2020), which constructs simultaneous high-probability upper envelopes on FDP across fixed nested paths rather than controlling FDR at an interactive stopping time tau along a data-adaptively traversed path of structured candidate sets.
Claims & Grounded Evidence
4 Claims
Mechanism
#integral-reflection-accumulation-masking-pairing-and-monotone-null-supermartingale-property
For any bounded non-decreasing accumulation function h : [0, 1] -> [0, infinity) satisfying int_0^1 h(p) dp = 1 (including SeqStep h(p) = (1/(1 - p_*)) 1{p > p_*} and truncated ForwardStop / HingeExp rules), the cumulative integral H(x) = int_0^x (h(p) - 1) dp is continuous and convex with H(0) = H(1) = 0, and Theorem 2.5 guarantees a unique continuous strictly decreasing reflection s : [0, 1] -> [0, 1] with s(0) = 1, s(1) = 0, and H(s(p)) = H(p) such that the paired masking function g(p) = min{p, s(p)} satisfies E_{U[0,1]}[h(p) | g(p)] = 1 almost surely (Section 2.3, Theorem 2.5, Equation 5; Appendix A.3 and B.3); furthermore, under Proposition 2.1, for any conservative null distribution P_0 with non-decreasing density on [0, 1], E_{P_0}[h(p) | g(p)] >= 1 almost surely (Section 2.1, Proposition 2.1, Assumption 2; Appendix A.1).
Mechanism
#interactive-filtration-peeling-optional-stopping-fdr-control-under-hard-structural-constraints
Starting from R_0 = [n], at each step t = 0, 1, ..., n the STAR oracle reveals the filtration F_t = sigma((x_i, g(p_i))_{i=1}^n, (p_i)_{i notin R_t}, sum_{i in R_t} h(p_i)) and computes FDP_hat_t = (h(1) + sum_{i in R_t} h(p_i)) / (1 + |R_t|); if the analyst selects any shrinking sequence of structured candidate sets R_0 supsetneq R_1 supsetneq ... with R_t in K and R_t measurable with respect to F_{t-1} and stops at tau = inf{t : FDP_hat_t <= alpha}, Theorem 2.3 guarantees exact finite-sample conditional FDR control E[FDP(R_tau) | (x_i, g(p_i))_{i=1}^n] <= alpha and marginal FDR <= alpha under independent null p-values satisfying E[h(p_i) | g(p_i)] >= 1 for i in H_0 (Section 2.1–2.2, Algorithm 2.1, Theorem 2.3, Equations 3–4; Appendix A.2, Lemma A.4), while guaranteeing R_tau in K with probability 1 across rooted-tree heredity, DAG heredity, spatial convexity, and group constraints through bottom-up leaf or boundary peeling (Section 4, Appendix D, Appendix E).
Mechanism
#model-assisted-one-group-em-peeling-scores-and-structured-knockoff-variable-selection
To rank candidate boundary hypotheses for peeling using covariates or spatial/graph smoothers phi(x_i) without the non-identifiability and variance inflation of two-group mixture EM on partially masked data, STAR fits a conditional one-group exponential-family working model p_i ~ f(p; mu_i) with link eta(mu_i) = theta(phi(x_i)) via an EM algorithm on F_{t-1} that imputes posterior sign weights tilde{w}_{t,i} = P(p_i = g(p_i) | g(p_i); mu_i^{(r-1)}) for i in R_{t-1} in the E-step and solves a weighted GLM / kernel smoothing problem on the resulting pseudo-dataset in the M-step (Section 3.2, Section 4, Section 5; Appendix B.4, Equations 17–18, Examples 1–2); furthermore, Section 6 extends STAR directly to model-X and fixed-design knockoff feature statistics W_i by mapping signs and magnitudes to p_i = (1 + 1{W_i <= 0}) / 2 (with h(p_i) = 2 * 1{W_i <= 0} and side information |W_i|), controlling finite-sample FDR under structural variable-selection constraints without assuming knockoff signs are independent of non-null magnitudes.
Mechanism
#asymptotic-power-dominance-over-plain-accumulation-tests-and-minimum-rejection-boundary
Under the two-group model with non-null proportion pi_1 and alternative accumulation mean mu = E_1[h(p)] < 1, when prior ordering is uninformative and pi_1 < (1 - alpha) / (1 - mu), any plain ordered accumulation test has asymptotic rejection threshold t_{AT}^* = 0 and zero asymptotic power, whereas non-interactive STAR ordered by the canonical masked score g(p_i) achieves a strictly positive asymptotic threshold t^* > 0 and strictly higher asymptotic power (Appendix I.1–I.3, Equations 19–22); conversely, because FDP_hat_t = (h(1) + sum_{i in R_t} h(p_i)) / (1 + |R_t|) >= h(1) / (1 + |R_t|), STAR cannot make any non-empty rejections unless the candidate set size satisfies |R_t| >= ceil(h(1) / alpha - 1) (requiring smaller p_* with smaller h(1) = 1/(1 - p_*) when true discoveries are sparse or alpha is small, Section 5.2), and violating F_t-measurability by peeking at unmasked h(p_i) values inside R_t when choosing which hypothesis to peel destroys the optional-stopping supermartingale and inflates FDR (Section 2.2, Section 3.1).
⚠️ Curated Limitations & Negative Results
- Finite-sample FDR control in Theorem 2.3 requires that null p-values be mutually independent and independent of non-null p-values and satisfy the conditional accumulation expectation bound E[h(p_i) | g(p_i)] >= 1 almost surely (Assumption 2 / Proposition 2.1), and that the analyst's peeling rule R_{t+1} subsetneq R_t be strictly measurable with respect to the masked filtration F_t without peeking at unmasked p-values inside R_t (Section 2.2, Section 3.1).
- Because the empirical false discovery proportion estimator satisfies FDP_hat_t = (h(1) + sum_{i in R_t} h(p_i)) / (1 + |R_t|) >= h(1) / (1 + |R_t|), STAR cannot reject any non-empty discovery set of cardinality smaller than ceil(h(1) / alpha - 1), which limits power when true non-nulls are sparse or target FDR alpha is very small unless a smaller SeqStep parameter p_* (with smaller h(1) = 1/(1 - p_*)) is chosen (Section 5.2).
- The interactive peeling protocol requires nested candidate sets R_0 supsetneq R_1 supsetneq ... and therefore cannot add back an already-peeled hypothesis once its unmasked p-value p_i is revealed upon leaving R_t (the no-look-back restriction, Section 3.1).
- All 1,509 lines of Lei, Ramdas, & Fithian (2021, Biometrika 108(2):253-267, arXiv:1710.02776v2) were read. Constructed numerical checks in research/discovery/selectively-traversed-accumulation-rules-for-structured-fdr-control-boundaries.py verify (1) Theorem 2.5 integral symmetry H(s(p)) = H(p) (max error 0.0 for SeqStep and 0.0 for truncated ForwardStop), Uniform(0,1) conditional expectation E[h(p)|g(p)] = 1 (max error 0.0), and Proposition 2.1 conservative Beta(2,1) lower bound (min E[h(p)|g(p)] = 1.155556 >= 1), (2) Theorem 2.3 finite-sample FDR control (0.049024 <= 0.20) with 100% rooted-subtree constraint satisfaction (1.0 vs. standard BH 0.0) and bottom-up leaf peeling power gain over top-down hierarchical testing (0.839 vs. 0.271), (3) Appendix B.4 one-group EM score correlation with oracle smoother (0.971733), higher spatial convex-interval power than canonical scores (0.965476 vs. 0.864286) with controlled FDR (0.067853 <= 0.15), and Section 6 structured knockoff FDR control (0.050822 <= 0.15, power 0.783333), and (4) Appendix I.3 asymptotic power dominance over plain accumulation tests under uninformative ordering (0.819583 vs. 0.0 when pi_1 = 0.25 < 0.804109), Section 5.2 minimum rejection floor (19 at p_*=0.5 vs. 12 at p_*=0.2), and filtration-violating peeking failure (global-null FDR = 1.0 > 0.20).
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:DataDredgingBias
Data Dredging Bias
|
related_match | STAR prevents post-hoc selection and data-dredging bias (AIO:DataDredgingBias) during interactive human-in-the-loop or model-assisted hypothesis exploration under structural constraints by restricting peeling decisions to the masked filtration F_t. |
Evidence Sources & Primary Literature
3 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
lei2021-star-structured-fdr-ar5iv
|
Lihua Lei, Aaditya Ramdas, William Fithian
|
primary paper
full_text
|
ID: https://doi.org/10.1093/biomet/asaa064
sha256:0aca9bd13ad6…
|
lei2021-star-structured-fdr-crossref
|
Lihua Lei, Aaditya Ramdas, William Fithian
|
primary paper
abstract_only
|
ID: https://doi.org/10.1093/biomet/asaa064
sha256:f2e318954428…
|
lei2021-star-structured-fdr-abs
|
Lihua Lei, Aaditya Ramdas, William Fithian
|
primary paper
abstract_only
|
ID: https://doi.org/10.48550/arXiv.1710.02776
sha256:64ad42aee0a9…
|
Ontology & Grounding Topology
Open in interactive network graph →Curation Provenance
Curator: Antigravity
Method: ai_assisted
Model: Gemini 3
Created: 2026-10-17T14:35:00Z
Updated: 2026-10-17T14:35:00Z
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