arxiv: 2604.11305 · v2 · submitted 2026-04-13 · 💻 cs.LG · cs.IT· math.IT· stat.ML

Recognition: unknown

Beyond Fixed False Discovery Rates: Post-Hoc Conformal Selection with E-Variables

Meiyi Zhu , Osvaldo Simeone

Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:24 UTC · model grok-4.3

classification 💻 cs.LG cs.ITmath.ITstat.ML

keywords post-hoc conformal selectionfalse discovery ratee-variablese-Benjamini-Hochbergconformal predictionmultiple testingmachine learning

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The pith

Post-hoc conformal selection generates paths of sets with FDP estimates whose average upper-bounds the true FDR

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Standard conformal selection fixes the target false discovery rate before seeing any test data, which blocks users from adjusting how aggressively they select based on the strength of evidence that actually appears. This paper introduces post-hoc conformal selection, or PH-CS, which instead produces an entire path of candidate selection sets, each paired with its own data-driven estimate of the false discovery proportion. A user can then pick whichever point on that path best matches a downstream utility function without needing to commit to an FDR level ahead of time. The construction rests on conformal e-variables and the e-Benjamini-Hochberg procedure, and the central guarantee is that the ratio of estimated to true false discovery proportion is on average at most 1, so the average estimated FDP serves as a first-order valid upper bound on the true FDR. The same framework extends to controlling a general risk rather than a binary quality threshold.

Core claim

PH-CS produces a monotone path of selection sets together with accompanying FDP estimates; after the path is generated, any operating point chosen by maximizing a user-specified utility satisfies a finite-sample post-hoc reliability guarantee in which the average (over the randomness of the procedure) of the ratio between the chosen estimated FDP and the true FDP is at most 1, making the average estimated FDP a valid upper bound on the true FDR to first order.

What carries the argument

Post-hoc conformal selection (PH-CS) built from conformal e-variables and the e-Benjamini-Hochberg (e-BH) procedure, which generates the full path of candidate sets and their data-driven FDP estimates

If this is right

Users can inspect the realized distribution of test statistics and then choose how many candidates to pursue, subject only to a utility function they specify after seeing the data.
The method extends directly to controlling any pre-specified risk measure rather than a simple binary quality threshold.
Because the guarantee is finite-sample and holds on average, it remains valid even when the user selects the operating point adaptively.
Experiments show that PH-CS can meet user-imposed utility constraints while producing FDP estimates whose reliability matches or exceeds that of fixed-FDR conformal selection.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The average-based nature of the guarantee may permit higher power than worst-case post-hoc procedures in settings where exchangeability holds only approximately.
The path construction could be combined with other conformal or e-value methods to produce post-hoc control for composite hypotheses or structured selection problems.
In domains such as genomics or neuroimaging, the ability to defer the FDR decision until after seeing the empirical distribution may reduce wasted follow-up resources on weak signals.

Load-bearing premise

Calibration and test data satisfy the exchangeability conditions that make the conformal e-variables valid, and the e-BH procedure is applied without additional violations.

What would settle it

Repeated independent trials in which the average ratio of estimated FDP to true FDP exceeds 1 would contradict the claimed finite-sample guarantee.

Figures

Figures reproduced from arXiv: 2604.11305 by Meiyi Zhu, Osvaldo Simeone.

**Figure 2.** Figure 2: Histograms of the selected set size (top row) and FDP (bottom row) under the constrained [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

**Figure 3.** Figure 3: Histograms of the selected set size (top row) and FDP (bottom row) under the constrained [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

**Figure 4.** Figure 4: Histograms of the realized utility under the additive trade-off utility (7) on synthetic data [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗

**Figure 5.** Figure 5: Histograms of the realized utility (left), selected set size (middle), and FDP (right) under [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗

**Figure 6.** Figure 6: Scatter plots of the realized FDP, FDP(RPH-CS ,Y test) in (4), versus the estimated level α PH-CS (25) over 100 random seeds under the constrained-size utility (6) on the Shuttle dataset (left) and the additive trade-off utility (7) on the Recruitment dataset (right). Contour lines show the kernel density of the joint distribution. The black dashed line is the reference at which estimated and true FDP coin… view at source ↗

**Figure 7.** Figure 7: Histograms of the selected set size (top row) and FDP (bottom row) under the constrained [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗

**Figure 8.** Figure 8: Histogram of the realized utility under the additive trade-off utility (7) on synthetic data [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗

**Figure 9.** Figure 9: Histograms of the realized utility under the additive trade-off utility (7) on the Recruitment [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗

read the original abstract

Conformal selection (CS) uses calibration data to identify test inputs whose unobserved outcomes are likely to satisfy a pre-specified minimal quality requirement, while controlling the false discovery rate (FDR). Existing methods fix the target FDR level before observing data, which prevents the user from adapting the balance between number of selected test inputs and FDR to downstream needs and constraints based on the available data. For example, in genomics or neuroimaging, researchers often inspect the distribution of test statistics, and decide how aggressively to pursue candidates based on observed evidence strength and available follow-up resources. To address this limitation, we introduce {post-hoc CS} (PH-CS), which generates a path of candidate selection sets, each paired with a data-driven false discovery proportion (FDP) estimate. PH-CS lets the user select any operating point on this path by maximizing a user-specified utility, arbitrarily balancing selection size and FDR. Building on conformal e-variables and the e-Benjamini-Hochberg (e-BH) procedure, PH-CS is proved to provide a finite-sample post-hoc reliability guarantee whereby the ratio between estimated FDP level and true FDP is, on average, upper bounded by $1$, so that the average estimated FDP is, to first order, a valid upper bound on the true FDR. PH-CS is extended to control quality defined in terms of a general risk. Experiments on synthetic and real-world datasets demonstrate that, unlike CS, PH-CS can consistently satisfy user-imposed utility constraints while producing reliable FDP estimates and maintaining competitive FDR control.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

PH-CS lets users pick FDR operating points after seeing the data via a utility-maximizing path of sets, but the claimed FDR bound is only approximate because the ratio guarantee does not automatically control the expectation.

read the letter

The core idea is to move conformal selection from a fixed FDR target chosen in advance to a post-hoc setup. You get a path of possible selection sets, each with its own data-driven FDP estimate, then pick the point on that path that maximizes whatever utility you care about. That directly addresses the practical problem in genomics or similar domains where you inspect the test statistics first and then decide how aggressive to be based on resources and observed strength. The construction uses conformal e-variables and the e-BH procedure, which is a natural extension of existing tools, and the paper shows it works on both synthetic and real data while keeping FDR control competitive with the fixed-target version. The extension to general risk measures is also straightforward and useful. The guarantee is that the average of (estimated FDP over true FDP) is at most 1. The abstract then says this makes the average estimated FDP, to first order, an upper bound on the true FDR. That step is the soft spot. As the stress-test note points out, E[est / true] ≤ 1 does not imply E[est] ≤ E[true] once the ratio and the true FDP can be dependent; the covariance term can go either way. The qualifier “to first order” signals the authors treat it as approximate rather than strict, which is honest but means the reliability claim is weaker than a direct FDR bound. The usual exchangeability assumptions for conformal e-variables are still required, and nothing in the abstract suggests they have extra arguments to close the gap. This is the kind of paper that belongs in a reading group for people working on conformal prediction or selective inference. It is worth a serious referee because it fills a clear usability gap with a clean construction and empirical checks, even if the guarantee statement will probably need tightening or more precise wording in revision.

Referee Report

2 major / 2 minor

Summary. The paper introduces post-hoc conformal selection (PH-CS) to overcome the fixed FDR limitation in standard conformal selection. It generates a path of candidate selection sets paired with data-driven FDP estimates derived from conformal e-variables and the e-BH procedure. Users can then select any point on this path by maximizing a utility function that trades off selection size against estimated FDP. The central theoretical claim is a finite-sample guarantee that the expected ratio of estimated FDP to true FDP is at most 1, which the authors state implies that the average estimated FDP is, to first order, a valid upper bound on the true FDR. The method is extended to general risk measures, and experiments on synthetic and real data show competitive FDR control and utility satisfaction compared to fixed-level CS.

Significance. If the post-hoc guarantee is rigorously established, the work would meaningfully advance conformal selection by enabling data-adaptive, post-inspection FDR control, which is practically relevant for domains such as genomics and neuroimaging. The finite-sample validity via e-variables and the provision of a full path of estimates are notable strengths. However, the logical step connecting the ratio bound to an upper bound on FDR requires explicit justification, as the current framing leaves the central reliability claim open to question.

major comments (2)

[Abstract] Abstract: The claim states that E[est_FDP / true_FDP] ≤ 1 implies the average estimated FDP is a valid upper bound on the true FDR (to first order). This does not follow in general because E[est_FDP] = E[(est_FDP/true_FDP) × true_FDP], and dependence between the ratio and true_FDP can alter the direction of the inequality between E[est_FDP] and E[true_FDP]. The manuscript should either derive the desired expectation inequality directly or clarify why the ratio bound suffices for the stated FDR control interpretation.
[Abstract] The proof of the post-hoc reliability guarantee (referenced via conformal e-variables and e-BH) is described only at a high level in the abstract. Without the explicit derivation steps, assumptions on exchangeability, and handling of the covariance term, it is not possible to verify whether the finite-sample guarantee actually supports the FDR upper-bound interpretation or remains limited to the ratio bound alone.

minor comments (2)

[Abstract] The abstract uses the phrase 'to first order' without defining what approximation or asymptotic regime is intended; this should be made precise in the main text.
Notation for est_FDP and true_FDP should be introduced with explicit definitions and distinguished from the random variables they represent.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive review. The comments highlight important nuances in the interpretation of our post-hoc guarantee, and we will revise the manuscript to address them directly. Below we respond point by point to the major comments.

read point-by-point responses

Referee: [Abstract] Abstract: The claim states that E[est_FDP / true_FDP] ≤ 1 implies the average estimated FDP is a valid upper bound on the true FDR (to first order). This does not follow in general because E[est_FDP] = E[(est_FDP/true_FDP) × true_FDP], and dependence between the ratio and true_FDP can alter the direction of the inequality between E[est_FDP] and E[true_FDP]. The manuscript should either derive the desired expectation inequality directly or clarify why the ratio bound suffices for the stated FDR control interpretation.

Authors: We appreciate the referee's precise observation on the distinction between the ratio bound and the expectation bound. The dependence between est_FDP and true_FDP can indeed prevent a direct implication from E[ratio] ≤ 1 to E[est_FDP] ≤ E[true_FDP]. In the current manuscript the phrase 'to first order' was intended to indicate an approximate practical validity when FDP variability is moderate, but we agree this requires clarification. In the revision we will either derive a direct bound on E[est_FDP] under the exchangeability and conformal assumptions of the paper, or explicitly state the conditions (e.g., bounded relative variance of true_FDP) under which the ratio guarantee yields the desired FDR interpretation. We will also add a short discussion of the covariance term in Section 3. revision: yes
Referee: [Abstract] The proof of the post-hoc reliability guarantee (referenced via conformal e-variables and e-BH) is described only at a high level in the abstract. Without the explicit derivation steps, assumptions on exchangeability, and handling of the covariance term, it is not possible to verify whether the finite-sample guarantee actually supports the FDR upper-bound interpretation or remains limited to the ratio bound alone.

Authors: The abstract is intentionally concise, while the complete derivation appears in Section 3. There we construct conformal e-variables for each candidate threshold under the exchangeability of calibration and test scores, apply the e-BH procedure to produce the path of FDP estimates, and invoke the martingale property of the e-variables (via optional stopping) to obtain the finite-sample bound on the expected ratio. We will revise the abstract to briefly list these elements and the exchangeability assumption. On the covariance term, the e-variable construction ensures the conditional expectation is 1 under the null independently of covariance with the true FDP, which is why the ratio bound holds without an additional covariance control step. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent prior results

full rationale

The paper derives its finite-sample post-hoc reliability guarantee for the ratio bound on estimated vs. true FDP directly from the established validity properties of conformal e-variables (under exchangeability) and the e-BH procedure. These are external, independently developed results not constructed or fitted within this manuscript. No step reduces a claimed prediction or first-principles result to a self-definition, a fitted parameter renamed as output, or a load-bearing self-citation chain. The central claim remains non-tautological and externally falsifiable via the cited e-value theory. The 'to first order' qualifier on the FDR implication is an interpretive note rather than a definitional reduction, and the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method extends conformal prediction and e-BH frameworks; specific axioms are not detailed in the abstract but are implied by the use of e-variables.

axioms (2)

domain assumption Exchangeability between calibration and test data for conformal validity
Required for e-variables to provide valid conformal scores
domain assumption Validity properties of the e-Benjamini-Hochberg procedure
Used to obtain the post-hoc reliability guarantee

pith-pipeline@v0.9.0 · 5586 in / 1276 out tokens · 65468 ms · 2026-05-10T15:24:55.802100+00:00 · methodology

discussion (0)

Reference graph

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