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Recertifying decisions via e-closure: uniform improvements of conformal risk control and randomized e-BH

Tian Bai, Aaditya Ramdas

arXiv:2610.35821v1stat.MEmath.STstat.MLstat.TH

TL;DR

Statistical procedures can leave part of their expected-loss budgets unused, motivating a way to improve decisions without losing their guarantees. The paper recertifies procedures by enlarging valid e-value certificates using computable slack. Across conformal risk control and randomized multiple testing, it reports less conservative or more powerful procedures, with application-specific costs and scope boundaries.

  • Problem

    Statistical procedures may leave risk budget unused even while controlling expected loss at a prescribed level, leaving room to improve their decisions under the same guarantee.

    Source: §1 · p. 1

  • Method

    Recertification measures unused budget in a valid procedure’s e-value certificate, enlarges the certificate while preserving validity, and inverts it into a less conservative decision rule.

    Source: §1 · p. 2, §2.3 · p. 7

  • Results

    CRC recertification is valid under every finite exchangeable law and no more conservative than CRC, while RU-eBH and RC-eBH improve randomized e-BH rejection sets and show greater power than e-BH and U-eBH in simulations.

    Source: §1 · pp. 2–3

  • Takeaways & Limitations

    The applications show that computable unused budget can support pointwise CRC improvements and closed-form randomized multiple-testing procedures with larger rejection sets.

    Source: §5 · p. 24

    Repeated CRC recertification trades computation for uniform improvement and can be expensive for score-dependent controllers; whether RC-eBH is admissible among all FDR procedures remains open.

    Source: §5 · p. 24

Abstract

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Statistical procedures often control expected loss at a prescribed level while leaving part of their risk budget unused. Building on the e-value representation of risk-controlled decisions and the e-closure principle for multiple testing, we formulate recertification, a general paradigm for improving procedures by exploiting computable slack in their validity certificates. A valid procedure's losses normalized by their risk budgets form a state-indexed e-value collection; conversely, decisions whose normalized losses are bounded by this collection inherit its risk guarantees. Recertification uses distributional structure to quantify slack, enlarge the certifying e-values while preserving validity, and construct less conservative decisions. We develop two applications. For conformal risk control (CRC) with a known loss family, recertification redistributes unused risk budget among the held-out points of each hypothetical augmented sample. The resulting controller is valid under every finite exchangeable law and no more conservative than CRC on any calibration sample. For convex losses on an interval of actions, repeated recertification converges geometrically to an admissible controller. For false discovery rate control with arbitrarily dependent e-values, recertification exploits unused conditional budgets in randomized e-BH. It yields a closed-form randomized procedure whose rejection set contains that of randomized e-BH for every realization of the shared randomization and whose expected number of rejections is maximal subject to the same conditional budget constraints. Recertifying with the larger e-collection of closed e-BH gives a second closed-form procedure whose rejection set contains that of the first. Experiments on CIFAR-100 hierarchical classification and multiple testing simulations demonstrate improved budget utilization, informativeness and power over the baselines.

1 Introduction

The paper frames statistical decisions as actions subject to statewise expected-loss budgets, then uses computable slack in validity certificates to recertify procedures without changing their guarantees. It develops this approach for conformal risk control and randomized multiple testing.

1 Introduction

  • Recertification enlarges a valid procedure’s canonical e-values using computable slack, then inverts the certificate to admit potentially less conservative actions with the same guarantee.The approach generalizes the e-value representation of risk-controlled decisions and can be repeated.Source: §1 · p. 2
  • For a fixed loss family and observation domain, CRC recertification redistributes unused budget across held-out positions, remains valid under every finite exchangeable law, and is no more conservative than CRC on any calibration sample.For convex losses, iterations converge geometrically to a controller admitting no further pointwise improvement.Source: §1 · p. 2
  • 17.2% fewer labels: on CIFAR-100, repeated recertification reduced mean prediction-set size from 10.77 to 8.92 while raising mean test risk from CRC’s 14.21% to 14.95% at a 15% target.Figure 1 summarizes risk-budget use and mean set size across 300 paired splits, each with 16 test images.Source: §1 · p. 2, Figure 1 · p. 3
  • RU-eBH spends unused conditional budget in randomized e-BH, rejecting at least as many hypotheses for every shared-randomization realization and maximizing expected rejections within the same certificate family.The procedure treats the conditional law of the rejection set given the e-values as a randomized action.Source: §1 · p. 3
  • Recertifying the larger closed e-BH e-value family yields RC-eBH, whose rejection set contains RU-eBH’s; simulations report both recertified rules more powerful than e-BH and U-eBH in every setting.RC-eBH was the most powerful of eight compared procedures in nearly every setting.Source: §1 · p. 3

1.1 Related work

  • Related work connects the paper to e-value decision certificates, closure principles, randomized multiple testing, and CRC extensions, while distinguishing recertification’s focus on improving existing procedures.The paper notes that representation and admissibility are separate questions, and that its CRC admissibility results rely on that problem’s structure.Source: §1.1 · pp. 3–4

1.2 Organization

  • The paper proceeds from general e-closure and recertification to CRC theory and real-data results, then randomized e-BH and simulations, before concluding.Source: §1.2 · p. 4

2 Generalized e-closure and recertification

The section extends e-closure from testing to general decision problems with state-indexed losses and budgets. It establishes a two-way correspondence between valid menus and e-value certificates, then uses conditional unused budget to enlarge those certificates while retaining validity.

2 Generalized e-closure and recertification

  • Generalized e-closure makes the correspondence between valid decisions and e-value certificates constructive, providing the framework for recertifying existing procedures.The section treats states, actions, and losses before establishing the correspondence and introducing recertification.Source: §2 · p. 4

2.1 States, actions, and a family of risks

  • States may specify classes of data distributions, actions may be arbitrary, and each state-loss pair has its own expected-loss budget, with losses potentially depending on the observation.The state, action, and loss-index sets may be uncountable; validity is required for every state, distribution, and loss.Source: §2.1 · p. 5

2.2 From e-collections to decisions and back

  • An e-collection certifies actions whose losses stay below the corresponding budget scaled by its e-value; conversely, every valid menu is certified by its canonical e-collection.Every measurable selection from a certified menu remains valid, and the resulting closure preserves the menu’s loss envelopes.Source: §2.2 · pp. 5–6
  • Randomized actions are probability distributions over actions, and their averaged losses can be certified even when their individual possible realizations are not.The guarantee averages over the subsequent random draw and does not cover choices made after that draw is observed.Source: §2.2 · pp. 6–7

2.3 Recertification: spending the unused budget

  • Recertification enlarges the loss envelopes within conditionally quantified unused budget, then closes the menu, retaining the original actions while certifying additional ones.The increments may depend arbitrarily on data but must respect conditional-expectation constraints; exchangeability supplies CRC’s allowance, while randomized e-BH uses known conditional losses given the e-values.Source: §2.3 · p. 7

3 Improving conformal risk control

This section develops recertification for conformal risk control, using unused risk budget on hypothetical augmented samples to improve controllers while preserving validity under every finite exchangeable law. It presents an exact account of the slack and one-step, iterative, and extended constructions, then applies the method to hierarchical image classification and discusses comparisons with CRC and envelope CRC and computational trade-offs.

3.1 Setup and two valid starting rules

  • CRC chooses actions to control expected test loss under finite exchangeability, while envelope CRC and standard CRC are valid baselines and envelope CRC is no more conservative when U ≤ B.Actions are ordered by conservativeness, and the goal is to be as liberal as possible subject to the risk guarantee.Source: §3.1 · p. 8

3.2 The exact risk budget on a hypothetical augmented sample

  • A symmetric controller is valid under every finite exchangeable law exactly when its held-out losses sum to at most (n + 1)α on every augmented multiset.The resulting nonnegative slack is the exact unused budget and is computable once the augmented sample is specified.Source: §3.2 · p. 9

3.3 A baseline-preserving refinement

  • Recertification keeps every valid symmetric controller valid while selecting actions no more conservative than the baseline on any calibration sample.It distributes each augmented sample’s unused budget across held-out positions and certifies the least conservative feasible action; equal sharing is one valid allocation.Source: §3.3 · pp. 9–10

3.4 Iteration to admissibility under convexity

  • With convex losses on an action interval, repeated recertification from any valid baseline decreases to a valid pointwise-admissible controller at geometric rate ρ = n/(n + 1).A valid controller is admissible exactly when it is a fixed point of recertification; convexity is needed for the admissible-limit result, not for validity of finite rounds.Source: §3.4 · p. 11

3.5 Extensions and computation

  • Recertification extends to jointly controlling arbitrarily many risks, preserving validity and baseline improvement, with the same admissible-limit rate under convex losses on an interval.The rate and admissibility conclusion do not depend on the number of risks.Source: §3.5 · p. 13
  • For monotone nonconvex losses, a one-pass sweep over calibration multisets yields a no-larger pointwise-admissible controller when the observation space is finite.The sweep uses only monotonicity and takes one pass over the controller table.Source: §3.5 · p. 13
  • Controllers may also use observed test features while retaining marginal risk guarantees, and one recertification round evaluates the baseline on nK replacement samples.Replacement recalibrations can share work; the number of rounds is a computational choice because validity holds after any number of rounds.Source: §3.5 · pp. 13–14
  • A data-independent baseline permits a reusable finite table, but individual features require separate replacement-sample tracking and continuum constraints need certified optimization or uniform approximation.Stopping by a fixed maximum round preserves validity, while conservative lower bounds on allowances preserve validity and baseline comparison but may prevent admissibility; runtimes range from about 0.2 seconds per common-depth table to 3–140 milliseconds per adaptive test image.Source: §3.5 · p. 14, §3.6.2 · p. 17

3.6.1 Hierarchical classification

  • Recertification is evaluated against ordinary CRC and envelope CRC using common-depth and confidence-adaptive rules for hierarchical classification.The classifier predicts among root, superclass, and leaf nodes; confidence-adaptive rules assign more specific predictions to images with higher classifier confidence.Source: §3.6 · p. 14, §3.6.1 · p. 15

3.6.2 Results

  • At α = 0.10, 500-round recertification raised confidence-adaptive correct specificity from 67.57% to 68.24% and reduced mean node size from 19.40 to 17.03 labels.At α = 0.02, the gain was 1.71 percentage points. In common-depth prediction, recertification added at most 0.44 percentage points over envelope CRC, while envelope CRC coincided with CRC for confidence-adaptive scores.Source: Table 2 · p. 16, §3.6.2 · p. 16
  • The first 50 rounds recovered at least 95% of the final gain in every setting, and later rounds changed little.At α = 0.10, one round gained 0.073 percentage points for common depth and 0.085 for confidence-adaptive scores, compared with 0.337 and 0.663 after 500 rounds.Source: §3.6.2 · p. 16, Figure 2 · p. 17
  • Common-depth predictions leave little recoverable slack after envelope CRC, whereas confidence-adaptive recertification uses unused budget even when the envelope equals the worst-case bound.Envelope CRC leaves about 2%–4% of the target unused for common depth; CRC leaves 5%–39% unused for confidence-adaptive scores, where gains shrink at larger targets as more baseline predictions are already leaves.Source: §3.6.2 · p. 17

4 Improving randomized e-BH

Recertification treats randomized rejection rules as actions certified by e-value budgets, then spends unused conditional budget to improve e-BH procedures. RU-eBH and RC-eBH retain FDR control under arbitrary dependence, with RC-eBH drawing on the larger closed e-BH e-collection.

4 Improving randomized e-BH

  • The section applies recertification to randomized e-BH with arbitrarily dependent e-values, treating the unknown true-null set as the state and false discovery proportion as the loss.The procedures are compared with six other methods that control FDR under arbitrary dependence.Source: §4 · p. 17, §4.5 · p. 22

4.1 e-BH and self-consistency

  • For e-BH, certification reduces to K singleton constraints, so its certified rejection sets are exactly the self-consistent sets and e-BH is the largest one.The condition is e_j ≥ K/(α|R|) for every j in R.Source: §4.1 · p. 18

4.2 Randomized e-BH as generalized e-closure

  • Randomized e-BH can be represented as a distribution over rejection counts, with validity requiring each hypothesis's expected charge to stay within its budget.U-eBH's common uniform factor can leave budget unused because it does not respect hypotheses' different slack structures.Source: §4.2 · pp. 18–19

4.3 Recertifying U-eBH

  • RU-eBH controls FDR at most α|T|/K, contains U-eBH and e-BH for every e and U, and is stochastically largest among boosting rules certified by e-BH's e-collection.Its probability of rejecting at least one hypothesis rises from (α/K) max_k k e[k] to (α/K) sum_j e_j, capped at one.Source: §4.3 · p. 20

4.4 Recertifying closed e-BH

  • RC-eBH uses the larger closed e-BH e-collection and contains RU-eBH's rejection set for every e and U, while maximizing expected rejections among rules certified by that collection.The procedure derives its rejection probabilities from the greatest convex minorant of cumulative budget caps.Source: §4.4 · p. 21

4.5 Simulations

  • With 10 false nulls, RU-eBH and RC-eBH lead every panel; at signal mean 2.5 with independent statistics, RU-eBH has power 0.32 versus 0.22 for U-eBH and J-eBH.RU-eBH's FDR is close to α|T|/K, while U-eBH reaches about 40%–70% of that bound; J-eBH has similar FDR to RU-eBH but lower power.Source: §4.5 · p. 22
  • With 90 false nulls, RC-eBH reaches power 0.64 versus 0.52 for RU-eBH and 0.55 for closed e-BH; it leads all eight methods except by less than 0.005 at the strongest independent signals.RC-eBH is more powerful than RU-eBH in every setting, while its FDR stays below α and can exceed α|T|/K when many hypotheses are false.Source: §4.5 · p. 22, Figure 3 · p. 23

5 Discussion

The discussion frames recertification as expanding a menu of valid actions and choosing among them. It distinguishes the paper's admissibility comparisons from classical risk admissibility and notes computational costs and an open admissibility question for RC-eBH.

5 Discussion

  • Recertification enlarges the certified menu, and the paper selects its least conservative controller or the rule rejecting the most hypotheses.Other data-dependent selections, including utility-maximizing ones, remain valid; admissibility comparisons here are samplewise or conditional, not classical risk admissibility.Source: §5 · p. 24
  • Repeated CRC recertification can be expensive when controllers depend on individual scores, while RU-eBH and RC-eBH require sorting and RC-eBH also requires a convex minorant.Whether RC-eBH is admissible among all FDR procedures remains open.Source: §5 · p. 24
  • More broadly, procedures with identifiable unused validity budgets are candidates for recertification, especially when their state, action, or loss structures are complex.The paper's framework formulates e-closure for general decision problems and uses the correspondence constructively.Source: §5 · p. 24

A Measurability of the certified menu

The appendix gives sufficient regularity conditions for the certified menu to support measurable selection, including utility-maximizing selection.

A Measurability of the certified menu

  • When the certified menu is nonempty for every observation, the stated regularity conditions yield a Borel graph, nonempty compact values, and a Borel selection.The conditions include standard Borel data space, separable metric state and loss-index spaces, compact metric actions, and finite losses and e-collection.Source: A Measurability of the certified menu · p. 28
  • Borel utilities continuous in the action admit a Borel maximizing selection from the menu.The loss envelopes are Borel in the observation and continuous in the state and loss index.Source: A Measurability of the certified menu · p. 28
  • The proof reduces constraints to a countable dense set using continuity, then applies the Arsenin–Kunugui theorem to obtain measurable selections.The same projection argument establishes measurability of maximizing selections and loss envelopes.Source: A Measurability of the certified menu · p. 28

B.1 CRC through e-closure

The appendix connects CRC to e-closure through losses on hypothetical augmented samples and shows how unused budget can be allocated among held-out positions. Pointwise controller improvements can be represented by such allocations, and repeated equal sharing converges to an admissible limit.

B.1 CRC through e-closure

  • Under the augmented-sample formulation, a fixed action's held-out losses define conformal e-values, and actions satisfying the samplewise loss budget belong to the certified menu.The resulting validity argument relies on exchangeability; finite exchangeable laws need not be mixtures of i.i.d. laws.Source: §B.1 · pp. 28–29
  • The conformal construction permits action-dependent e-values because, on each augmented sample, whether an action passes is determined by the state rather than which point is held out.The sharp envelope CRC test follows by bounding the unseen test loss with the sharp envelope; larger envelopes or standard CRC's constant bound make the test more stringent.Source: §B.1 · p. 29

B.2 Other allocations

  • Permutation-equivariant weights allocate unused budget across held-out positions, and every pointwise improvement between jointly valid symmetric controllers admits such an allocation.Equal weights recover the paper's equal-sharing construction; weights proportional to held-out losses are another option.Source: §B.2 · pp. 29–30
  • Repeated equal sharing converges to an admissible controller that satisfies the allocation condition, and no allocation yields a valid controller that improves on that limit.Source: §B.2 · p. 30

B.3 Proof of Theorem 3.5 and Proposition 3.6

For convex losses on an interval, the recertification operator yields a jointly valid controller that is no more conservative than any jointly valid controller below it. Its iterates converge geometrically to an admissible controller.

B.3 Proof of Theorem 3.5 and Proposition 3.6

  • If A and C are jointly valid with C ≤ A, then the recertified controller satisfies RA ≤ ρA + C/(n + 1).The proof bounds each held-out loss increase using the unused budget and applies convexity to an interpolated action.Source: §B.3 · p. 30
  • The iterates remain jointly valid, decrease pointwise, and converge to a jointly valid limit by right-continuity of the losses.The limit is admissible, and its gap from the kth iterate contracts by factor ρ at each step.Source: §B.3 · p. 30
  • A jointly valid controller is admissible exactly when it is a fixed point of the recertification operator.If A is admissible, RA is valid and no larger than A; conversely, the one-step comparison rules out any strictly smaller valid controller when RA = A.Source: §B.3 · p. 30

B.4 Proof of Proposition 3.7

For a finite observation space, a predetermined sweep lowers controller entries whenever the joint-validity constraints permit. The resulting controller is jointly valid, no larger than the original, and admissible.

B.4 Proof of Proposition 3.7

  • Changing one controller entry affects only augmented multisets formed by adding a held-out point to that calibration multiset.For repeated observations, the changed entry appears once for each matching held-out position.Source: §B.4 · p. 31
  • A valid controller can be strictly improved exactly when some single entry can be lowered while satisfying its local constraint (3.13).The argument compares any pointwise smaller valid controller with a change at one calibration multiset.Source: §B.4 · p. 31
  • A predetermined sweep over the finitely many calibration multisets lowers each entry to the smallest action satisfying its validity constraint.Each update preserves joint validity, and later updates cannot make a visited entry reducible again.Source: §B.4 · p. 31

B.5 The two-type example

In the two-type example, recertification from envelope CRC converges geometrically to a limit controller. The example also shows that admissible controllers can be incomparable and that the interval-action assumption matters.

B.5 The two-type example

  • Starting from envelope CRC, the update raises controller entries subject to the unused-budget constraints at augmented multisets.The example specifies linear validity constraints for HH, HHL, HLL, and LLL; envelope CRC has entries (1/2, 3/5, 3/4).Source: §B.5 · pp. 31–32
  • ρ = 2/3, and the iterates converge to A(∞) = (1/2, 3/8, 1/8).The coordinates are the controller entries at HH, HL, and LL.Source: §B.5 · p. 32
  • The limit and a second valid controller are both admissible, but neither dominates the other; their pointwise minimum is invalid.The limit has (h, u, v) = (1/2, 5/8, 7/8), while the second controller has (h, u, v) = (1/2, 3/5, 9/10).Source: §B.5 · p. 32
  • Theorem 3.5 requires an interval of actions: on a grid, its restricted update can stop at a fixed point strictly dominated by a valid controller.The sweep of Proposition 3.7 avoids interpolation and reaches the dominating controller in one pass.Source: §B.5 · p. 32

B.6 A two-risk example with a continuous observation space

A two-risk example with continuous observations shows recertification improving on joint envelope CRC. Its iterates converge geometrically to the largest valid controller dominating that baseline.

B.6 A two-risk example with a continuous observation space

  • The joint envelope baseline takes the more conservative of the two bounds, while A* is characterized by the tighter pointwise limit.The baseline is A0(x) = 1/[2(1 + max(x, 1 − x))], and A*(x) = [1 + min(x, 1 − x)]/4.Source: §B.6 · p. 33
  • The limiting controller A* is the largest valid controller dominating the joint envelope baseline A0, with strict improvement at every interior point.The example uses two losses, ya and (1 − y)a, each with risk budget 1/4 under every exchangeable law of (X, Y).Source: §B.6 · pp. 32–33
  • The iterates increase to A*, with geometric rate ρ = 1/2 attained with equality.Each iterate is a convex combination of A0 and A*, so validity follows from the linear constraints.Source: §B.6 · p. 33

B.7 Implementation details for the real-data study

The real-data study implements recertification through held-out loss budgets and evaluates its outputs on shared CIFAR-100 splits. In the randomized e-BH appendix, U-eBH leaves rank-specific budget slack that defines a larger certified menu.

B.7 Implementation details for the real-data study

  • For common depth, the update chooses the largest next depth whose type-specific loss stays within its current loss plus an equal share of unused budget.The update uses the two possible nonzero label types and has the closed form in (B.6).Source: §B.7 · p. 34
  • Common-depth recertification uses a data-independent table indexed by calibration type counts, with 1,326 entries at n = 50.Each round updates the whole table, which can be computed once per target and reused across splits.Source: §B.7 · p. 34
  • Confidence-adaptive scores require a per-image table with r2^(r−1) entries for r active points, making computation more demanding than common depth.Capping breakpoints and grouping identical breakpoint pairs reduce table size without changing depths.Source: §B.7 · pp. 34–35
  • Larger nominal levels leave more distinct breakpoints after capping, explaining the longer runtimes.The number of allowable jumps rises with the level, raising the capping threshold τ.Source: §B.7 · p. 35
  • Each real-data split uses 50 calibration and 16 test images sampled without replacement, with all methods evaluated on the same 300 splits.Metrics are computed exactly over prediction randomization and averaged across test images and splits.Source: §B.7 · p. 35

C.1 The certified menu and U-eBH

  • U-eBH leaves positive rank-specific budget slack when e[r] > e[k_r], and recertification makes the full certified menu available.Every distribution in the menu satisfies the stated charge budgets; the menu contains the original U-eBH distribution.Source: §C.1 · p. 36

C.2 Proof of Proposition 4.1

The proof constructs a distribution in the certified menu whose rejection-count tails attain the largest allowable bounds. Under shared randomization, its rejection set contains U-eBH's, and its decisions can be represented by valid e-values.

C.2 Proof of Proposition 4.1

  • When a1 ≤ 1, the constructed distribution is untruncated and every constraint in the certified menu holds with equality.In this case, p*_k = ep_k for k ≥ 1.Source: §C.2 · p. 36
  • The constructed distribution is a measurable selection from the certified menu and controls FDR at level α.Its rejection-count tails attain the maximum bounds among rules selecting from that menu.Source: §C.2 · p. 36
  • With the same randomization, the constructed rule's rejection set contains U-eBH's, which in turn contains e-BH's.Its tails are at least U-eBH's at every rank, while every certified-menu rule has tails bounded by the constructed rule's tails.Source: §C.2 · p. 36
  • The rule's rejection set can be recovered by applying e-BH to derived e-values whose conditional expectations are bounded by the original e-values.The derived e-values assign K/(α|R|) to rejected hypotheses and zero to the others.Source: §C.2 · p. 37

C.3 Randomization over arbitrary rejection sets

The appendix considers randomized rules over arbitrary rejection sets, not only nested rejection sets. Under the same certified budgets, the constructed rule maximizes expected rejections, although other certified rules can have larger rejection-tail probabilities at some ranks.

C.3 Randomization over arbitrary rejection sets

  • For arbitrary rejection-set distributions, EeBH certification is equivalent to each hypothesis's expected false-discovery share staying within its budget.The share charged to hypothesis j is denoted c_j(Q), with budget b_j = (α/K)e_j.Source: §C.3 · p. 37
  • J-eBH is certified by EeBH and therefore controls FDR despite using non-nested rejection sets.The proof bounds each hypothesis's charge by conditioning on the other uniforms and integrating over its own uniform.Source: §C.3 · p. 37
  • Every distribution over rejection sets certified by EeBH has expected rejections no greater than the constructed rule, but need not obey its tail bounds.The appendix gives a certified example with Q(|R| ≥ 2) = 0.078 versus q*_2 = 0.05, while its expected rejection count is 1.08 versus 1.2.Source: §C.3 · pp. 37–38
  • A dual certificate for the budget-constrained linear program establishes the expected-rejection bound.The program maximizes E_Q|R| subject to the per-hypothesis charge constraints.Source: §C.3 · pp. 37–38

C.4 Proof of Proposition 4.2

The proof constructs a second randomized rule from the closed e-BH e-collection. Its rejection set contains the rule from Proposition 4.1's construction, and the proof establishes validity and optimality within monotone rules.

C.4 Proof of Proposition 4.2

  • The constructed rejection-count distribution has nonnegative, non-increasing charges and is a measurable selection from the certified menu.Its tails are q♯_r, and its rule is valid for every nonempty true-null set.Source: §C.4 · pp. 38–39
  • The proof reduces validity checks to states containing the hypotheses with the smallest e-values.For each fixed state size, exchanging a selected rank for a larger rank does not increase the constraint expression, so the largest-e-value state is the binding check.Source: §C.4 · p. 39
  • The closed e-BH rule is valid and, under shared randomization, its rejection set contains that of the first recertified rule.Its rejection set is itself certified by EeBH.Source: §C.4 · pp. 39–40
  • Among monotone rules certified by EeBH, the constructed rule maximizes expected rejections.Its distribution attains the greatest convex minorant S, which bounds the corresponding sequence for any feasible distribution.Source: §C.4 · pp. 38–39

C.5 The minimally adaptive improvement

The minimally adaptive rule R† preserves FDR guarantees while containing RU-eBH's rejection set and sometimes improving it. The conditional-calibration discussion interprets e-BH recalibration as spending discarded evidence using a known conditional law.

C.5 The minimally adaptive improvement

  • R† has FDR at most α in every state T and under every law in P_T.Source: §C.5 · p. 40
  • R† contains RU-eBH's rejection set for every E and U, and under some law makes strictly more true discoveries with positive probability.Source: §C.5 · p. 40
  • Because R† is certified by EeBH, RC-eBH rejects at least as many hypotheses as R† in expectation given T.Source: §C.5 · p. 40
  • With probability α/(2K), R† makes K−1 more true discoveries than RU-eBH in the stated law and e-value example.For 1/2 < U ≤ (K + 1)/(2K), RU-eBH rejects only H1 while R† rejects all K hypotheses.Source: §C.5 · p. 40

C.6 Conditional calibration as recertification

  • The method assumes Sj makes X's conditional law known and common under Hj; when its budget condition holds, recalibrated e-values remain valid and e-BH rejects at least R0.Hypotheses outside R0 are added when their inflated e-values reach K/(α(r + 1)).Source: §C.6 · p. 41
  • Conditional calibration restates Lee and Ren's method as recertification, spending e-BH-pruned evidence across data values sharing Sj rather than across external-seed values.Source: §C.6 · p. 41
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