Simultaneous pathwise false discovery proportion bounds
Mathematical Methods AIMech:SimultaneousPathwiseFalseDiscoveryProportionBounds DRAFT
Aliases: Simultaneous High-Probability FDP Bounds Across Nested Rejection Paths, Closed-Testing Interpolation-Based Simultaneous FDP Control, Pathwise False Discovery Proportion Confidence Envelopes, Simultaneous FDP Bounds for Sorted, Interactive, Knockoff, and Online Paths
A finite-sample multiple-testing and selective-inference methodology (Katsevich & Ramdas 2020, The Annals of Statistics 48(6):3465-3487, arXiv:1803.06790) that constructs simultaneously valid level-(1 - alpha) upper confidence envelopes on the false discovery proportion FDP(R_k) = V(R_k) / max(1, |R_k|) across an entire nested sequence of candidate rejection sets emptyset = R_0 subset R_1 subset ... subset R_m ordered by sorted p-values, covariate-assisted interactive unmasking (STAR/AdaPT), knockoff feature-importance statistics, or online hypothesis streams. By proving a unifying finite-sample exponential supermartingale and empirical-process inequality (Lemma 1 and Lemma 2) to obtain pathwise false-discovery upper bounds V(R_k) <= bar{V}(R_k) and applying closed-testing interpolation tilde{V}(R_k) = min_{0 <= j <= k} {bar{V}(R_j) + |R_k \ R_j|} (Proposition 1), the procedure allows analysts to inspect the entire size-versus-FDP trade-off curve and select any rejection set R_k or target FDP level post hoc without violating simultaneous coverage.
Identity Notes & Concept Scope
No normalized identity match for simultaneous pathwise false discovery proportion bounds across all 442 pinned AIO records and 584 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches across data, sources, research, history, and reviews returned only references in AdaptivePValueThresholding, FocusedBH, and ConformalNoveltyDetection context notes. Distinct from AIMech:AdaptivePValueThresholding (Lei & Fithian 2018, AdaPT) and AIMech:FocusedBH (Katsevich, Sabatti, & Bogomolov 2023), which output a single stopping threshold controlling expected FDR = E[FDP] <= q rather than simultaneous high-probability upper bounds on FDP(R_k) across all path steps k; distinct from AIMech:GeneralizedAlphaInvesting, AIMech:LOND, AIMech:LORD, and AIMech:SAFFRON, which control online mFDR or expected FDR for a single sequence of rejection decisions rather than simultaneous post-hoc FDP envelopes across all candidate online sets (Theorems 4-5); and distinct from AIMech:DerandomizedKnockoffs (Ren, Wei, & Candes 2023), which aggregates multiple knockoff runs for single-threshold PFER or k-FWER control rather than bounding FDP simultaneously across the entire ordered knockoff path (Theorem 3).
Claims & Grounded Evidence
4 Claims
Mechanism
#closed-testing-interpolation-and-sorted-p-value-poissonized-empirical-process-bound
For any nested path of candidate rejection sets emptyset = R_0 subset R_1 subset ... subset R_m equipped with any simultaneous level-(1 - alpha) false-discovery upper bound V(R_k) <= bar{V}(R_k) satisfying P(forall k, V(R_k) <= bar{V}(R_k)) >= 1 - alpha, the closed-testing interpolation bound tilde{V}(R_k) = min_{0 <= j <= k} {bar{V}(R_j) + |R_k \ R_j|} and tilde{FDP}(R_k) = tilde{V}(R_k) / max(1, |R_k|) satisfy P(forall k, V(R_k) <= tilde{V}(R_k)) >= 1 - alpha with tilde{V}(R_k) <= min(|R_k|, bar{V}(R_k)) and tilde{FDP}(R_k) <= 1 (Section 2.1, Proposition 1, Equation 7); for the canonical sorted p-value path R_k = {i : p_i <= p_{(k)}} under mutually independent super-uniform null p-values (Condition 1), setting hat{V}_{sort}(R_k) = n_0 p_{(k)} <= n p_{(k)} yields the simultaneous level-(1 - alpha) bound bar{V}_{sort}(R_k) = floor(c(alpha) max(1, hat{V}_{sort}(R_k))) with closed-form multiplier c(alpha) = inf_{x >= max(1, sqrt(log(1/alpha)/2 - 1/8) + sqrt(1/8))} 2 x^2 / (2 x sqrt(log(1/alpha)) + log(1/alpha)) > 1 (Section 2.2, Theorem 1, Equations 8–9; Section 2.3, Proposition 2), derived via Poissonization and the sharp one-sided Poisson supremum inequality P(sup_{t >= 0} N(t) / max(1, t) >= c) <= exp(-u_c^2 / 2) where u_c = c / sqrt(2(c - 1)) - 1 (Section 7.1, Lemma 2).
Mechanism
#unifying-exponential-supermartingale-lemma-and-knockoff-simultaneous-fdp-bound
Given a filtration (F_k)_{k>=0} and F_k-measurable nondecreasing processes (V_k, A_k)_{k>=0} with V_0 = A_0 = 0, binary false-discovery increments Delta V_{k+1} in {0, 1}, and predictable offsets o_{k+1} in [0, 1] satisfying Delta A_{k+1} >= o_{k+1} + (1 - o_{k+1}) Delta V_{k+1} and E[exp(theta Delta V_{k+1} - psi(theta) Delta A_{k+1}) | F_k] <= 1 with psi(theta) = log(1 + theta) / (1 + theta) for theta = log c > 0, the process M_k = exp(theta V_k - psi(theta) A_k) is a nonnegative F_k-supermartingale with M_0 = 1 and Ville's maximal inequality guarantees P(forall k >= 0, V_k <= floor(c(alpha) + (log c / c)(c A_k - V_k))) >= 1 - alpha for c(alpha) = log(1/alpha) / log(1 + log(1/alpha)) (Section 7.2, Lemma 1); specializing to model-X or fixed-X knockoff paths R_k = {j : |W_j| >= |W|_{(k)}, W_j > 0} under i.i.d. coin-flip null sign symmetry (Condition 2, o_{k+1} = 1/2), Theorem 3 establishes that bar{V}_{knock}(R_k) = floor(c(alpha) + c(alpha) hat{V}_{knock}(R_k)) with hat{V}_{knock}(R_k) = |{j : |W_j| >= |W|_{(k)}, W_j < 0}| and tighter constant c(alpha) = log(1/alpha) / log(2 - alpha) is simultaneously valid across all k (giving c(0.05) = 4.485775 versus 1/alpha = 20 for backward-supermartingale knockoff bounds, Section 4, Equations 15–18).
Mechanism
#interactive-masked-p-value-peeling-simultaneous-fdp-envelopes
In interactive and structured multiple testing with side information X_i or geometric/DAG constraints (STAR / AdaPT framework), decomposing each p-value into a masked value tilde{p}_i = min(p_i, 1 - p_i) and sign s_i = 1(p_i <= 1/2) allows a human or automated analyst to iteratively peel hypotheses along a shrinking nested sequence [n] = R_0 supset R_1 supset ... supset R_m = emptyset adapted to the masked filtration F_{k-1} = sigma((X_i)_{i=1}^n, (tilde{p}_i)_{i=1}^n, (p_i)_{i notin R_{k-1}}, |{i in R_{k-1} : p_i <= 1/2}|); under independent mirror-conservative null p-values (Condition 1, P(p_i <= 1/2 | tilde{p}_i) <= 1/2), applying Lemma 1 to the empirical false-discovery estimator hat{V}_{int}(R_k) = |{i in R_k : p_i > 1/2}| yields the finite-sample simultaneous bound bar{V}_{int}(R_k) = floor(c(alpha) + c(alpha) hat{V}_{int}(R_k)) with c(alpha) = log(1/alpha) / log(1 + log(1/alpha)) across all peeling steps k (Section 3, Theorem 2, Equations 12–14), achieving substantially higher discovery power than covariate-ignorant sorted p-value bounds when covariates are informative (Section 6.2).
Mechanism
#online-conditional-superuniformity-bounds-and-offline-independence-boundary
For online hypothesis streams (H_1, H_2, ...) where predictable target levels alpha_j in (0, 1) and optional screening thresholds lambda_j in [0, 1) are measurable wrt F_{j-1} and null p-values satisfy conditional super-uniformity P(p_j <= u | F_{j-1}) <= u (Condition 3), rejecting R_k = {j <= k : p_j <= alpha_j} admits both the simple online simultaneous bound bar{V}_{online}(R_k) = floor(c(alpha) + c(alpha) sum_{j <= k, j in H_0} alpha_j) <= floor(c(alpha) + c(alpha) sum_{j=1}^k alpha_j) (Section 5, Theorem 4, Equations 19–22) and the adaptive online bound bar{V}_{adapt-online}(R_k) = floor(c(alpha) + c(alpha) sum_{j=1}^k (alpha_j / (1 - lambda_j)) 1(p_j > lambda_j)) with c(alpha) = log(1/alpha) / log(1 + log(1/alpha)) simultaneously across all times k >= 1 (Theorem 5, Equations 23–25); by contrast, the offline sorted p-value bound bar{V}_{sort} (Theorem 1) requires mutual independence across null p-values (Condition 1) and can exceed its nominal failure rate alpha under strong positive equicorrelation across offline tests.
⚠️ Curated Limitations & Negative Results
- Theorem 1 (sorted p-value path) requires mutual independence among null p-values (Condition 1); under strong positive equicorrelation across offline tests, the supremum ratio sup_k V(R_k) / max(1, n_0 p_{(k)}) becomes heavy-tailed and can violate the nominal level alpha unless dependence-robust or resampling-based bounds are substituted.
- Theorem 2 (interactive covariate-assisted path) and Theorem 3 (knockoff path) require independent mirror-conservative null p-values (P(p_i <= 1/2 | tilde{p}_i) <= 1/2) or exact i.i.d. coin-flip null sign symmetry (Condition 2); misspecification of the feature distribution in model-X knockoffs or asymmetric null p-values can invalidate the exponential supermartingale inequality of Lemma 1.
- Because the bounds hold simultaneously across all path steps k and with high probability 1 - alpha over the realized FDP rather than merely in expectation (FDR), the multiplicative constants c(alpha) > 1 (e.g., c_sort(0.10) = 1.927324 and c_knock(0.05) = 4.485775) and additive offset c(alpha) introduce conservatism in very small rejection sets compared to single-threshold expected-FDR methods when only a fixed pre-specified FDR target is needed.
- All 1,158 lines of Katsevich & Ramdas (2020, The Annals of Statistics 48(6):3465-3487, arXiv:1803.06790v5) were read. Constructed numerical checks in research/discovery/simultaneous-pathwise-false-discovery-proportion-bounds-boundaries.py verify (1) Section 2.2 Theorem 1 / Section 7.1 Lemma 2 exact multiplier c_sort(0.10) = 1.927324, Proposition 2 threshold cutoff alpha(1.5) = 0.318532, tighter intermediate bound V_bar_sort = 26 vs. DKW = 66 and Robbins = 125, and global-null simultaneous violation rate 0.056667 <= 0.10 vs. unadjusted 1.0, (2) Section 7.2 Lemma 1 exact supermartingale increment inequality <= 1.0 across Cases 1-3, Section 4 Theorem 3 knockoff constant c_knock(0.05) = 4.485775 vs. backward 20.0, and knockoff simultaneous violation rate 0.006 <= 0.05, (3) Section 3 Theorem 2 interactive covariate-assisted path simultaneous violation rate 0.0225 <= 0.10 and higher average post-hoc discoveries (32.335 vs. 24.8425 at tilde{FDP} <= 0.20) than sorted p-values, and (4) Section 5 Theorems 4-5 online simple and adaptive simultaneous bounds (violation rates 0.001 <= 0.10) vs. offline violation rate 0.23 > 0.10 under strong positive equicorrelation rho = 0.90.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:DataDredgingBias
Data Dredging Bias
|
related_match | Simultaneous pathwise FDP bounds and closed-testing interpolation guarantee P(forall k, FDP(R_k) <= tilde{FDP}(R_k)) >= 1 - alpha across the entire rejection path, preventing post-hoc threshold-snooping and data-dredging inflation (AIO:DataDredgingBias). |
AIO:TimeSeriesAnalysis
Time Series Analysis
|
related_match | Theorems 4 and 5 construct simultaneous pathwise FDP bounds across online sequential hypothesis streams (H_1, H_2, ...) adapted to a filtration (F_k)_{k>=0} under conditional super-uniformity (AIO:TimeSeriesAnalysis). |
Evidence Sources & Primary Literature
3 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
katsevich2020-simultaneous-fdp-ar5iv
|
Eugene Katsevich, Aaditya Ramdas
|
primary paper
full_text
|
ID: https://doi.org/10.1214/19-AOS1938
sha256:b636ac132952…
|
katsevich2020-simultaneous-fdp-crossref
|
Eugene Katsevich, Aaditya Ramdas
|
primary paper
abstract_only
|
ID: https://doi.org/10.1214/19-AOS1938
sha256:1a675bc830c8…
|
katsevich2020-simultaneous-fdp-abs
|
Eugene Katsevich, Aaditya Ramdas
|
primary paper
abstract_only
|
ID: https://doi.org/10.48550/arXiv.1803.06790
sha256:84e6c38e4fb7…
|
Ontology & Grounding Topology
Open in interactive network graph →Curation Provenance
Curator: Antigravity
Method: ai_assisted
Model: Gemini 3
Created: 2026-10-17T14:15:00Z
Updated: 2026-10-17T14:15:00Z
Source File:
data/curated/SimultaneousPathwiseFalseDiscoveryProportionBounds.yaml
· Schema:
curated_concept.yaml
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