Bounding Neyman-Pearson Region with $f$-Divergences

Andrew Mullhaupt; Cheng Peng

REVIEW 5 major objections 5 minor 26 references

Bounding Neyman-Pearson Region with $f$-Divergences

T0 review · 5 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For any f-divergence, the achievable false-positive/false-negative rates obey one convex inequality, and the bound is best possible.

desk verdict The lower-bound framework and realization results are worth a look, but Theorem 3's refined upper bound is false, and the paper needs major revision before its results can be used. read the letter →

arxiv 2505.08899 v1 pith:ELGZC2NF submitted 2025-05-13 math.ST cs.LGstat.MLstat.TH

Andrew Mullhaupt , Cheng Peng This is my paper · ORCID

classification math.STcs.LGstat.MLstat.TH MSC 62F0362B1060E15

keywords f-divergenceNeyman-PearsonregionROCcurveBayeserrorrateChernoffcoefficienthockey-stickdivergencePinskerinequalityhypothesistesting

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that the optimal trade-off between false positives and false negatives in testing two simple hypotheses is governed by every f-divergence: for any convex f, the achievable rates (α,β) satisfy (1−α) f(β/(1−α)) + α f((1−β)/α) ≤ D_f(P||Q). Because the hockey-stick family realizes all supporting lines of the Neyman-Pearson boundary, the bound cannot be improved for general f. Along the way the paper shows the KL version is stricter than Pinsker's inequality, gives a tensorized lower bound from α-divergences, produces a closed-form Chernoff-coefficient upper bound refined by the line of ignorance, and describes how to realize any boundary by categorical or interval distributions. A reader should care because this packages classical testing limits into a single inequality that captures the information an f-divergence carries about whether two distributions can be told apart.

What carries the argument

The engine is the perspective identity behind Jensen's inequality. For a nonrandomized test set E, the proof splits the f-divergence integral over E and its complement, applies Jensen to both pieces, and uses the extreme-point lemma to discard singular sets. The inequality's left side is the perspective function of f evaluated at the two error rates, which is convex. The tightness mechanism is the hockey-stick divergence f(t)=max{t−γ,0}: its piecewise-linear breakpoint makes Jensen's inequality an equality exactly when the test set is a likelihood-ratio threshold {p/q>γ}, so varying γ gives all supporting lines of the boundary. The upper-bound machinery is the Chernoff α-coefficient ρ_q=∫p^q $q^{{1−q}}$, whose tensorization and closed-form envelope give a power-law upper bound.

What would settle it

Pick a pair of distributions with a known Neyman-Pearson boundary, compute D_{f_γ} for many γ, and check whether each line β=−γα+1−D_{f_γ}(P||Q) touches the boundary at a likelihood-ratio threshold; a single γ with a gap would disprove the claimed tightness. Separately, evaluate the extreme-point inclusion (7) on an upper-boundary extreme point, where the inclusion pattern is reversed, to see that the lemma as stated is false.

Watch

Extended reading notes

Core claim

The central discovery is a convex lower bound on the Neyman-Pearson region in terms of any f-divergence D_f(P||Q): every achievable pair (α,β) satisfies (1−α) f(β/(1−α)) + α f((1−β)/α) ≤ D_f(P||Q). The left side is a convex function of (α,β), so the inequality describes a convex set containing the entire Neyman-Pearson region. The bound is sharp in a strong sense: choosing the hockey-stick divergence f(t)=max{t−γ,0} yields the family of lines β≥−γα+1−D_{f_γ}(P||Q), and every one of these lines is a supporting line to the boundary. Consequently no f-divergence bound can be tighter for all divergences. The paper also shows the KL case is tighter than Pinsker's inequality, and that the Chernoff α-coefficient gives a closed-form upper bound that can be refined by taking the convex envelope with the line of ignorance.

Load-bearing premise

The proof of the main inequality assumes that at an extreme point of the Neyman-Pearson region, the test set fully contains points on which only the first distribution has mass and fully excludes points on which only the second has mass; this inclusion pattern is asserted for all extreme points, though only the lower boundary, where the theorem is used, needs it.

Editorial extensions

If this is right

For KL divergence, the resulting inequality β ln(β/(1−α)) + (1−β) ln((1−β)/α) ≤ KL(P||Q) is strictly tighter than the Pinsker-derived bound α+β≥1−√(KL/2), and it remains nontrivial even when the Pinsker bound is vacuous.
For α-divergences and product measures, replacing ρ_q with the product of per-coordinate coefficients gives a tensorized lower bound, which translates directly into a lower bound on the sample size needed to reach a target error pair for i.i.d. observations.
The Chernoff α-coefficient yields a closed-form upper bound β ≤ (ρ_q q^q (1−q)^{1−q})^{1/q} α^{(q−1)/q}, and refining this with the line of ignorance gives a convex, piecewise-defined bound with two tangent lines.
Any convex Neyman-Pearson boundary can be realized exactly by a uniform distribution paired with a distribution whose quantile function is the inverse boundary, and any boundary can be approximated by a pair of categorical distributions with prescribed likelihood-ratio slopes.
Every supporting line to the Neyman-Pearson boundary is also a Bayes error line for some class probability, so the lower and upper bounds translate directly into bounds on the Bayes error rate under every prior.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

My inference: the tightness result reinterprets an f-divergence as a summary of the entire optimal error trade-off, so comparing two divergences with different f is less informative than comparing their full hockey-stick curves; the paper hints at this but does not state the comparison rule.
My inference: because every divergence estimate yields a valid lower bound on the Neyman-Pearson region, any consistent divergence estimator gives a practical pre-training certificate that a classifier cannot beat a given error pair; the paper develops the inequality but not the workflow.
My inference: the realization theorems suggest a reverse route for model criticism—given an empirical ROC curve, one can construct a distribution pair whose optimal boundary matches it and then use that pair as a benchmark null; the paper does not draw this benchmarking application.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a mixed bag. The main lower bound (Theorem 1, Eq. 8) is the data processing inequality for f-divergences applied to the test channel, not a new inequality, but the exposition is clean and the hockey-stick example showing that the f-divergence family cannot be improved is a nice observation. The closed-form Chernoff envelope (Proposition 3) and the explicit construction theorems (Theorems 4 and 5) are genuinely useful, and the tensorized lower bound has practical sample-size implications. Those contributions deserve credit.

The central problem is Theorem 3. The claimed refined upper bound takes the minimum of the hyperbola and two tangent lines. The tangents lie below the hyperbola, so taking their minimum can cut below the true Neyman-Pearson boundary. Concretely: let P=(0.8,0.2), Q=(0.2,0.8), q=1/2, rho=0.8. At alpha=0.01 the true boundary is beta=0.96 (randomized likelihood-ratio test), while Theorem 3 gives beta <= min{4, 0.984, 0.6336} = 0.6336, which is not an upper bound. The correct convex envelope is the maximum of the two tangents, not the minimum. This is a load-bearing flaw: the abstract and contributions advertise a closed-form refined upper bound, and the error appears in the main display (21) and its proof (A.16). The stress-test note holds up.

Two smaller issues are worth flagging. Lemma 1 is stated for all extreme points but the inclusion of singular sets is only valid on the lower boundary; the upper-boundary case has the inclusions reversed. The proof of Theorem 1 only needs the lower-boundary case, so the theorem survives, but the lemma overclaims. Proposition 5 also overreaches: randomizing a boundary test with the line-of-ignorance test only reaches points between those two curves, not arbitrary ROC curves. And the novelty claim for Theorem 1 should acknowledge that Eq. 8 is the data processing inequality in disguise.

Who is this for? People working on divergence-based bounds for classification will find useful examples and the realization trick, but they should not rely on Theorem 3 or Proposition 5 until corrected. A serious referee should see this paper because the valid parts are worth publishing after revision, but the false theorem and the overclaims must be fixed first. I would send it to peer review with a clear instruction to verify the convex envelope calculation and to correct Lemma 1 and Proposition 5.

Referee Report

5 major / 5 minor

Summary. The paper studies the Neyman-Pearson region for simple binary hypothesis testing, i.e., the set of achievable false-positive/false-negative pairs (α,β). Its central claim is a general lower bound, Theorem 1 (Eq. (8)): for any f-divergence D_f, every point of the Neyman-Pearson boundary satisfies (1−α) f(β/(1−α)) + α f((1−β)/α) ≤ D_f(P||Q). The authors argue this bound is best possible because the family of hockey-stick divergences yields all supporting lines to the Neyman-Pearson boundary. They then derive special cases for total variation, α-divergences, squared Hellinger distance, and KL divergence, and note a tensorization property for independent samples. Section 4 proposes an upper bound derived from the Chernoff α-coefficient and a refined closed-form upper bound in Theorem 3. Section 5 gives realizability results for arbitrary Neyman-Pearson boundaries, and Sections 6–7 connect the region to Bayes error rate and ROC curves.

Significance. If the main inequalities are correct, the paper provides a clean, parameter-free lower bound that unifies several known bounds and sharpens Pinsker-type statements in the KL case. The hockey-stick characterization of the Neyman-Pearson boundary is a nice optimality statement, and the tensorization observation for α-divergences is practically useful. The realizability constructions and the Bayes-error/ROC dictionary are also potentially valuable. However, the headline refined upper bound, Theorem 3, is false as stated, and several auxiliary lemmas and propositions overreach. The lower-bound part is defensible and worth publishing after correction, but the upper-bound and realizability sections need substantial rework.

major comments (5)

[Section 4, Eq. (21); proof in A.16] Theorem 3 is false as stated. A concrete counterexample is P=(0.8,0.2), Q=(0.2,0.8), with q=1/2. The Hellinger affinity is ρ_{1/2}=0.8. At α=0.01, the Neyman-Pearson boundary is β=0.96: the likelihood-ratio test includes state 1 with probability 0.05, giving Q-mass 0.01 and P-mass 0.04, so the false-negative rate is 0.96. The right-hand side of Eq. (21) is min{16, 0.9844, 0.6336}=0.6336, which is strictly below the true boundary. The line −ρ_q^{1/q}α+ρ_q^{1/q} is the tangent to the Chernoff upper-bound curve at α=1/2, but a tangent to an upper bound is not itself an upper bound on the whole interval; it falls below the true boundary for small α. The proof of A.16 treats the tangent lines as valid upper bounds globally. The 'lower convex envelope' in Theorem 2 must be the greatest convex minorant of min{g,h}, and Eq. (21) is not that object; the pointwise minimum of g and two tangents can lie below the Neyman-Pearson boundary. This is a load-bearing error for the paper's claimed closed-form refined upper bound.
[Section 3, Lemma 1; proof in A.1] Lemma 1 states that every extreme point of the Neyman-Pearson region satisfies the singular-set inclusions in Eq. (7), with {x:q=0,p>0} included in E and {x:q>0,p=0} excluded from E. This is true only for extreme points on the lower Neyman-Pearson boundary. For extreme points on the upper boundary, the inclusions are reversed: the likelihood-ratio test that realizes the upper boundary accepts H1 on low-likelihood-ratio points, so the q=0,p>0 set should be excluded and the p=0,q>0 set should be included. The proof in A.1 shows the claimed interiority only when β<1−α, which is exactly the lower-boundary case. Since Theorem 1 concerns the lower Neyman-Pearson boundary, the main lower bound survives, but the lemma as stated is false and should be restricted or corrected.
[Section 7, Proposition 5; proof in A.20] Proposition 5 claims that an arbitrary ROC curve g(t) can be realized by a one-parameter family of tests that randomize between a test on the Neyman-Pearson boundary and a test on the line of ignorance. This is too strong. A randomized mixture of the boundary test (with true positive rate 1−f(t)) and the line-of-ignorance test (with true positive rate t) can only produce ROC points whose true positive rate lies between t and 1−f(t). If g(t) lies below the line of ignorance or above the optimal ROC curve, the mixing weights in A.20 are outside [0,1] or the target point is unattainable. The proposition should be restricted to ROC curves that are pointwise between the line of ignorance and the optimal ROC curve.
[Section 5, Theorem 5; proof in A.17] The proof of Theorem 5 uses α_j=Σ_{i=1}^j p_i and β_j=1−Σ_{i=1}^j q_i for the j-th change point. This is inconsistent with the paper's definition (5), where α=Q(E) and β=P(E^c); for E consisting of the j largest likelihood ratios, one should have α_j=Σ_{i=1}^j q_i and β_j=1−Σ_{i=1}^j p_i. With the displayed assignments, the computed slope k_i=q_j/p_j is the reciprocal of the slope k_i=p_i/q_i stated in the theorem. This appears to be a transposition of P and Q in the proof, and it must be corrected for the realizability construction to be verifiable.
[Section 5, Lemma 2; proof in A.18] Lemma 2 asserts α(E)≤α((1−µ(E),1)) for any measurable E when the cdf F of Q is convex, where µ(E)=P(E). The proof in A.18, however, establishes only the opposite rearrangement inequality α((0,µ(E)))≤α(E) for the leftmost subset of P-measure µ(E). The stated rightmost inequality is not proved. The gap is likely fixable by applying the same rearrangement argument to the rightmost subset or to the complement, but as written the lemma is not supported by its proof, and Theorem 4 relies on it.

minor comments (5)

[Section 3, Example 4] The text says the squared Hellinger distance is a special case of 'Example 4' but it should refer to Example 3 (the α-divergence example).
[A.11] There are two subsection headings both labeled 'Proof of Example 6'; the second should be renumbered, and the equation numbering in that part skips from (96) to (97) with no corresponding (96) in the displayed block.
[Notation, Section 2] The symbol q is used both for the density of Q and for the parameter of the α-divergence and Chernoff coefficient. The authors warn about this, but the double use makes formulas such as ρ_q and q(x) unnecessarily easy to confuse; a different symbol for the parameter would improve readability.
[A.5, Proposition 1] The proof says the point (α,β) is 'the only intersection' between the supporting line and the Neyman-Pearson region. This is false when the boundary contains a linear segment, e.g., for categorical distributions with a density ratio equal to the threshold γ, where the line coincides with the boundary segment. The supporting-line conclusion is still correct, but the proof should be rephrased to say the region lies entirely on one side of the line.
[Abstract and title] The title has a missing space ('withf-Divergences'), and the abstract contains minor formatting issues. These should be corrected in revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No load-bearing circularity: Theorem 1 is a Jensen/data-processing derivation, the optimality claim is a supporting-line argument, and the only self-citation is a non-load-bearing recovery remark.

full rationale

I traced the claimed derivation chain and found no step that reduces to its own inputs. Theorem 1 (Eq. 8) is proved from Lemma 1 and Jensen's inequality; it is essentially the data-processing inequality for the binary test channel, but that is a derivation from stated assumptions, not an assumption of the target inequality. The optimality claim does not assume the boundary: Proposition 1 verifies equality for hockey-stick divergences at likelihood-ratio test sets, so the hockey-stick family yields supporting lines and hence a tight universal lower bound. No parameter is fitted to data and then renamed a prediction; the bounds are closed-form inequalities in the divergence values. The refined upper bound (Theorem 3) is obtained by envelope and tangent calculations from the Chernoff coefficient, so it is not circular, even if a tangent-line calculation were numerically wrong; mathematical error is a correctness issue, not circularity. The only self-citation, to Ding and Mullhaupt (2023), appears in Proposition 2 as a recovery statement ('This recovers the bound in Ding, Mullhaupt (2023)') and is not used to justify any premise of the main theorems. Realization results in Section 5 are explicit constructions from a prescribed boundary rather than imports of that boundary as an assumption. The paper is therefore self-contained against external benchmarks such as Pinsker's inequality, TVD bounds, and the Chernoff bound. No circular step is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted constants or invented entities. The paper's parameters (γ for hockey-stick, q for Chernoff coefficient) are indexing parameters of divergence families, not free parameters fitted to data. The derivations rely on standard convex analysis and the Neyman-Pearson lemma as background.

assumptions (4)

standard math Jensen's inequality for the probability measure q dλ/(1−α) on the rejection set E^c
Used in Eqs. (29)-(32) of the proof of Theorem 1 to pull f inside the integral.
domain assumption Neyman-Pearson fundamental lemma: the lower boundary of the feasible region is achieved by likelihood ratio tests
Invoked in Section 2 to define the Neyman-Pearson boundary and in Proposition 1 to identify the supporting lines.
domain assumption P and Q are absolutely continuous with respect to a common dominating measure λ
Needed for the f-divergence integral representation in Definition 1.
domain assumption Convexity of the Neyman-Pearson region and its vertical slices, giving a convex lower boundary B(α)
Used in Theorem 1 to extend the bound to randomized tests via convex hull, and in Theorem 4 to justify using B^{-1}(1−x) as a convex CDF.

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Pith. "Pith review of Bounding Neyman-Pearson Region with $f$-Divergences." pith.science (2026). https://pith.science/paper/ELGZC2NF

@misc{pith2026250508899,
  author       = {Pith},
  title        = {Pith review of: Bounding Neyman-Pearson Region with $f$-Divergences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ELGZC2NF}},
  note         = {Machine review of arXiv:2505.08899}
}

abstract

The Neyman-Pearson region of a simple binary hypothesis testing is the set of points whose coordinates represent the false positive rate and false negative rate of some test. The lower boundary of this region is given by the Neyman-Pearson lemma, and is up to a coordinate change, equivalent to the optimal ROC curve. We establish a novel lower bound for the boundary in terms of any $f$-divergence. Since the bound generated by hockey-stick $f$-divergences characterizes the Neyman-Pearson boundary, this bound is best possible. In the case of KL divergence, this bound improves Pinsker's inequality. Furthermore, we obtain a closed-form refined upper bound for the Neyman-Pearson boundary in terms of the Chernoff $\alpha$-coefficient. Finally, we present methods for constructing pairs of distributions that can approximately or exactly realize any given Neyman-Pearson boundary.

Figures

Figures reproduced from arXiv: 2505.08899 by the authors.

**Figure 2.** Refined closed-form upper bound for Neyman-Pearson boundary generated by Hellinger [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Supporting lines to Neyman-Pearson region corresponding to hockey-stick divergences with different γ [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Methods of information geometry

Amari Shun-ichi, Nagaoka Hiroshi . Methods of information geometry. 191. 2000

work page 2000

[2] [2]

Robust solutions of optimization problems affected by uncertain probabilities // Management Science

Ben-Tal Aharon, Den Hertog Dick, De Waegenaere Anja, Melenberg Bertrand, Rennen Gijs . Robust solutions of optimization problems affected by uncertain probabilities // Management Science. 2013. 59, 2. 341--357

work page 2013

[3] [3]

Empirically Estimable Classification Bounds Based on a Nonparametric Divergence Measure // IEEE Transactions on Signal Processing

Berisha Visar, Wisler Alan, Hero Alfred O., Spanias Andreas . Empirically Estimable Classification Bounds Based on a Nonparametric Divergence Measure // IEEE Transactions on Signal Processing. II 2016. 64, 3. 580--591

work page 2016

[4] [4]

On Stein’s lemma in hypotheses testing in general non-asymptotic case // Statistical Inference for Stochastic Processes

Burnashev Marat V . On Stein’s lemma in hypotheses testing in general non-asymptotic case // Statistical Inference for Stochastic Processes. 2023. 26, 1. 89--97

work page 2023

[5] [5]

Wasserstein upper bounds of the total variation for smooth densities // Statistics & Probability Letters

Chae Minwoo, Walker Stephen G . Wasserstein upper bounds of the total variation for smooth densities // Statistics & Probability Letters. 2020. 163. 108771

work page 2020

[6] [6]

A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations // Annals of Mathematical Statistics

Chernoff Herman . A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations // Annals of Mathematical Statistics. 1952. 23. 493--507

work page 1952

[7] [7]

Differential and Integral Calculus, Volume 2

Courant Richard . Differential and Integral Calculus, Volume 2. 2. 2011

work page 2011

[8] [8]

a t von Markoffschen Ketten // A Magyar Tudom \'a nyos Akad \'e mia Matematikai Kutat \'o Int \'e zet \'e nek K \

Csisz \'a r Imre . Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizit \"a t von Markoffschen Ketten // A Magyar Tudom \'a nyos Akad \'e mia Matematikai Kutat \'o Int \'e zet \'e nek K \"o zlem \'e nyei. 1963. 8, 1-2. 85--108

work page 1963