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arxiv: 2606.27349 · v2 · pith:KLREFKTOnew · submitted 2026-06-25 · 💻 cs.IT · math.IT· math.PR· math.ST· stat.ML· stat.TH

All you need is log

Akshay Balsubramani This is my paper

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classification 💻 cs.IT math.ITmath.PRmath.STstat.MLstat.TH
keywords multi-distribution divergencesRényi divergencesdata processing monotonicityproduct additivitycoincidence divergencesmulti-hypothesis testinginformation measures
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The pith

Every functional on W-tuples of distributions that is monotone under data processing and additive on independent products equals a positive integral of multi-way coincidence divergences over four strata.

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

The paper shows that the unique family of divergences for comparing any number of probability distributions at once is obtained by integrating the basic multi-way coincidence divergences C_α. These are defined by C_α(π1,…,πW) = -log ∫ π1^α1 ⋯ πW^αW where the exponents sum to one. The integral runs over a four-part parameter space whose strata each correspond to a distinct limiting regime that cannot be reproduced by the others. The same family emerges from axioms, entropy means, hypothesis-testing exponents, and a betting interpretation, recovering the ordinary Rényi divergences when only two distributions are compared.

Core claim

Every functional of W-tuples of distributions that is monotone under data processing and additive on independent products is a positive integral of the multi-way coincidence divergences C_α(π1,…,πW) := -log∫ π1^α1⋯πW^αW (∑αk=1) over a parameter space with four strata: the simplex interior; mixed-sign exponent cones; a tropical boundary at infinity carrying max-divergences; and pairwise Kullback-Leibler edges at the simplex vertices. Each stratum is necessary because it is the destination of an explicit data-processing-monotone, product-additive divergence that the others cannot reproduce, and each arises as a clean limit of simplex-interior atoms. The two-prior case recovers the standard Rén

What carries the argument

The multi-way coincidence divergences C_α(π1,…,πW) := -log∫ π1^α1⋯πW^αW (∑αk=1), which act as the generating atoms whose positive integrals over the four-stratum parameter space exhaust all functionals obeying the two structural axioms.

If this is right

  • When W equals 2 the construction reduces exactly to the classical Rényi family.
  • Multi-population fairness measures, multi-prior PAC-Bayes bounds, and multi-hypothesis testing error exponents are all instances of the same integral family.
  • The family can be derived from Kolmogorov-Nagumo means obeying Rényi entropy axioms or from a multi-lottery betting interpretation.
  • A conditional extension of the same integral representation holds for conditional distributions.

Where Pith is reading between the lines

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

  • The four-stratum decomposition may supply a systematic way to interpolate between different multi-distribution measures used in ensemble methods.
  • Numerical checks for small W could verify whether common ad-hoc multi-distribution scores already belong to the integral family.
  • The tropical boundary stratum suggests possible max-based approximations that remain monotone and additive.
  • Extending the characterization to continuous parameter spaces or to quantum states would test the robustness of the axiomatic route.

Load-bearing premise

The functional is required to satisfy monotonicity under data processing and additivity on independent products for arbitrary W-tuples.

What would settle it

Exhibit a concrete functional on three or more distributions that obeys both the data-processing monotonicity and independent-product additivity axioms yet lies outside every positive integral of the C_α over the four strata.

Figures

Figures reproduced from arXiv: 2606.27349 by Akshay Balsubramani.

Figure 1
Figure 1. Figure 1: Sym-orbit average vs closed-form symmetric-form representation (Section 7): the relative residual sits near machine precision (10−16) across all 1,200 trials, every one passing the 10−10 pre-registered threshold (dashed). G.2 Convergence-rate slopes for the KL and tropical limit identities The vertex-derivative identity (9) predicts |C(1−ϵ)ek+ϵel (π)/ϵ − D1(πkkπl)| = O(ϵ) as ϵ ↓ 0, i.e. slope +1 on a log-l… view at source ↗
Figure 2
Figure 2. Figure 2: Fitted log-log slope versus theoretical slope per(W, X) cell. KL stratum (blue) clusters tightly around +1 (theo￾retical rate); tropical stratum (vermillion) clusters around −0.97, just above the theoretical −1 due to the sub-exponential log(t)/t correction from the Laplace prefactor. above −1 (median fitted slope −0.95 to −0.98) and are consistent with the predicted slope once the prefactor correction is … view at source ↗
Figure 3
Figure 3. Figure 3: Per-(W, X) cell agreement rate between the spectrum inequality spec and the construction-flag cat (whether π ′ was drawn as Kπ). The dashed line marks the pre-registered 95% threshold; the pooled agreement rate is 95.6%, with no false negatives (zero “Kπ violates the inequality”) and a small number of false positives (random π ′ accidentally passing the sampled grid). G.4 Choquet linearity holds on every… view at source ↗
Figure 4
Figure 4. Figure 4: Choquet-linearity sweep across six cells of the atom-family cone. Per-cell pass rates for joint DPI, additivity, and ground state. All three axioms pass in every cell; the pre-registered 99% threshold (dashed) is exceeded uniformly. The structural reading: cone-additivity is genuinely cell-uniform, not C-specific. r ⋆ = p ⋆ α⋆ ∝ Q k π α ⋆ k k at the LHS argmax (the saddle-point form of Sion’s theorem), the… view at source ↗
Figure 5
Figure 5. Figure 5: Sion minimax identity residual vs grid resolution δ at W = 3. Median and maximum relative residual across 50 trials (10 per X ∈ {4, 5, 6, 8, 10}). A decade in δ yields roughly a decade in residual; V6’s coarser δ ≈ 1/30 residual of 2.4% is consistent with the extrapolation of this curve to δ = 3·10−2 . adult__additivity adult__ground_state adult__joint_dpi bank__additivity bank__ground_state bank__joint_dp… view at source ↗
Figure 6
Figure 6. Figure 6: Real-data axiom-stress on natural class-conditional distributions (UCI Adult, UCI Bank, MNIST, CIFAR-10, ImageNet-1K). Per-(dataset, axiom) Wilson 95% confidence interval on the passage rate at relative tolerance 10−6 . The dashed line marks the 99% pre-registered threshold. All fifteen cells achieve passage rate 1.0 with Wilson lower bound at least 0.99. 45 [PITH_FULL_IMAGE:figures/full_fig_p045_6.png] view at source ↗
read the original abstract

Comparing two probability distributions is a basic building block of statistics and machine learning, and the right family is well understood: the R\'enyi divergences of order $\alpha\in[0,\infty]$ are the unique family monotone under data processing and additive on independent products. Many problems instead compare more than two distributions at once -- multi-population fairness, multi-prior PAC-Bayes bounds, multi-hypothesis testing -- and the right multi-distribution generalization of the R\'enyi family has been an open question. We characterize it. Every functional of $W$-tuples of distributions that is monotone under data processing and additive on independent products is a positive integral of multi-way coincidence divergences $C_{\alpha}(\pi_1,\dots,\pi_W) := -\log\int \pi_1^{\alpha_1}\cdots\pi_W^{\alpha_W}$ (with $\sum_k \alpha_k = 1$) over a parameter space with four strata: the simplex interior; mixed-sign exponent cones (the analogue of R\'enyi orders $>1$); a tropical boundary at infinity carrying max-divergences; and pairwise Kullback-Leibler edges at the simplex vertices. Each stratum is necessary -- the destination of an explicit data-processing-monotone, product-additive divergence the others cannot reproduce -- and each is a clean limit of simplex-interior atoms. The same family arises from several independent routes -- the structural axioms, Kolmogorov-Nagumo means with R\'enyi's entropy axiomatics, classical entropy characterizations, multi-hypothesis testing error exponents, and a multi-lottery betting interpretation -- structural evidence that this is the canonical multi-distribution R\'enyi calculus rather than an artefact of any one axiomatic input. The two-prior case recovers the standard R\'enyi result; a worked $W=3$ instance, numerical verification, and a conditional extension round out the treatment.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript claims to characterize every functional of W-tuples of distributions that is monotone under data processing and additive on independent products as a positive integral of multi-way coincidence divergences C_α(π1,…,πW) := -log∫ π1^α1⋯πW^αW (∑αk=1) over a parameter space with four strata (simplex interior, mixed-sign exponent cones, tropical boundary at infinity, and pairwise KL edges). Each stratum is shown necessary via explicit constructions, the family arises from five independent routes (structural axioms, Kolmogorov-Nagumo means, entropy characterizations, testing error exponents, betting), the W=2 case recovers Rényi divergences, and a W=3 instance plus conditional extension are provided.

Significance. If the characterization holds, this supplies the canonical multi-distribution Rényi calculus, unifying several strands in information theory with direct applications to multi-population fairness, multi-prior PAC-Bayes, and multi-hypothesis testing. The five-route derivation and explicit necessity constructions constitute strong structural evidence; the recovery of the classical two-distribution case and the conditional extension are additional strengths.

minor comments (3)
  1. The abstract states that each stratum is a clean limit of simplex-interior atoms, but the precise limiting argument (including any required regularity on the measure) should be stated explicitly in the main text rather than left to the constructions.
  2. Notation for the four strata would benefit from a single summary table or diagram listing the parameter domains, the corresponding divergence forms, and the explicit constructions that demonstrate necessity.
  3. The numerical verification for the W=3 case should report the specific ranges of α vectors tested and the quantitative error metric used to confirm agreement with the integral representation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so there are no individual points requiring point-by-point rebuttal or revision.

Circularity Check

0 steps flagged

Multiple independent routes support the representation; no load-bearing circularity

full rationale

The paper characterizes all functionals of W-tuples satisfying monotonicity under data processing and additivity on independent products as positive integrals of the C_α family over four strata. This is derived directly from the stated axioms. The abstract explicitly lists five independent supporting routes (structural axioms, Kolmogorov-Nagumo means, classical entropy characterizations, hypothesis testing exponents, betting interpretation) and notes that the W=2 case recovers the known Rényi result. Each stratum is shown necessary via explicit constructions. No step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the result is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claim rests on the two structural axioms of data-processing monotonicity and product additivity; the coincidence divergence C_α is introduced as the atomic functional, with the four strata obtained as limits; no fitted numerical parameters appear.

axioms (2)
  • domain assumption Monotonicity under data processing for any W-tuple functional
    Invoked as the first defining property that forces the integral form (abstract, paragraph 2).
  • domain assumption Additivity on independent products for any W-tuple functional
    Invoked as the second defining property that forces the integral form (abstract, paragraph 2).
invented entities (1)
  • multi-way coincidence divergence C_α no independent evidence
    purpose: Atomic building block whose positive integrals over four strata yield all admissible functionals
    Defined directly as -log∫ π1^α1⋯πW^αW with ∑αk=1; no external falsifiable evidence supplied beyond the axiomatic derivation.

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