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Sharp informational inequalities involving Kullback-Leibler and R\'enyi divergences and a family of scaling-invariant relative Fisher measures

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Pith's one-line read This paper introduces a relative differential-escort transformation and uses it to prove sharp lower bounds, in terms of Kullback-Leibler and Rényi divergences, for two new families of scaling-invariant relative Fisher and…

desk verdict Worth engaging: the new scale-invariant relative Fisher and cumulative measures are genuinely new and mostly proven, but the Stam-like inequality has a real absolute-continuity gap on bounded supports that a referee should insist be patched. read the letter →

arxiv 2507.17408 v1 pith:5OX347PV submitted 2025-07-23 math-ph math.MP

Razvan Gabriel Iagar , David Puertas-Centeno , Elio V. Toranzo This is my paper · ORCID
classification math-phmath.MP MSC 94A1726D1560E15
keywords relativeFisherdivergenceRényiKullback-Leiblerdifferential-escorttransformationcumulativemomentstretchedGaussiangeneralizedtrigonometricfunctionsStaminequality
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

The paper's aim is to build a relative version of the differential-escort transformation, a change of variables plus rescaling of a density by a reference density $h$, and to use it to define two new biparametric families of informational measures: a relative Fisher divergence and a relative cumulative moment. The central achievement it asserts is that these measures are invariant under scaling of both densities, unlike previously known relative Fisher measures, and that they satisfy sharp lower bounds controlled by the Kullback-Leibler and Rényi divergences. Explicit minimizers are given as inverse relative differential-escort transforms of stretched Gaussians, and for a prescribed minimizer the reference density takes a closed form built from generalized trigonometric functions. The authors see this as transferring the classical moment-entropy and Stam inequalities from the single-density setting to the relative framework.

What carries the argument

The engine of the paper is the relative differential-escort transformation $R^{[h]}_\alpha[f](y) = (f(x)/h(x))^\alpha$, with the new coordinate defined by $y'(x)=f(x)^{1-\alpha}h(x)^\alpha$ (Definition 3.1). It generalizes the differential-escort transformation: choosing the reference density $h$ to be uniform on the support recovers the standard case. The transformation is a bijection between densities when the Rényi entropy power $N_{1/\alpha^*}[g]$ is finite, with inverse $R^{-1,[h]}_\alpha[g]$ built by scaling $g$ by its Rényi entropy power. The mechanism of the proofs is to compute, in terms of the original pair $(f,h)$, the Rényi entropy power (Lemma 3.2), the $(p,\lambda)$-Fisher information (Lemma 4.1) and the $p$-th cumulative moment of the transformed density, so that the classical inequalities (2.9) and (2.7) become the new sharp relative inequalities.

What would settle it

Take the reference $h$ to be a Gaussian and pick $f$ such that $f/h$ is smooth with unbounded derivative on an unbounded support; if the left side of (4.8) can be driven below the claimed constant $\alpha^{(\lambda-\beta-1)/(\alpha\beta)}(K^{(1)}_{p,\beta,\lambda})^{1/\alpha}$, then the absolute-continuity hypothesis is genuinely needed. Alternatively, for $h$ exponential and $f$ equal to the claimed minimizer $R^{-1,[h]}_\alpha[g_{p,\lambda}]$, evaluate both sides of (4.7) numerically; equality to machine precision would confirm the sharpness computation.

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Extended reading notes

Core claim

The central discovery is that the relative differential-escort transformation $R^{[h]}_\alpha[f](y)=(f(x)/h(x))^\alpha$ with $y'=f^{1-\alpha}h^\alpha$ converts a classical inequality for a single density into a sharp inequality for a pair of densities. Applying the classical moment-entropy inequality to $R^{[h]}_\alpha[f]$ yields $e^{D_{\xi(\lambda,\alpha)}[f\|h]}\sigma_{p^*,\alpha}[f\|h] \geq (K^{(0)}_{p,\lambda})^{1/\alpha}$ (Theorem 4.1), and applying the triparametric Stam inequality yields the companion bound $(e^{-D_{\xi(\lambda,\alpha)}[f\|h]}\phi_{p,\beta\alpha}[f\|h])^{1+\beta-\lambda} \geq \alpha^{(\lambda-\beta-1)/(\alpha\beta)}(K^{(1)}_{p,\beta,\lambda})^{1/\alpha}$; in both cases equality is attained by the inverse transform of the stretched Gaussian minimizers $g_{p,\lambda}$ and $g_{p,\beta,\lambda}$. Because the new relative Fisher divergence and relative cumulative moment are invariant under simultaneous scaling, the bounds are not an artifact of units, and the $\lambda\to 1$ case yields clean inequalities involving the Shannon entropy and the Kullback-Leibler divergence.

Load-bearing premise

The argument inherits the classical Stam inequality (2.7), which requires the parameter sign condition (2.6) and absolute continuity of the transformed density $R^{[h]}_\alpha[f]$; the admissible pairs $(f,h)$ satisfying these conditions are not fully characterized, and the sharpness claim also assumes $N_{1/\alpha^*}[g_{p,\lambda}]$ is finite.

Editorial extensions

If this is right

  • For any fixed reference density $h$, inequalities (4.7) and (4.8) give explicit, sharp lower bounds on the relative cumulative moment and relative Fisher divergence in terms of Rényi divergences, with the minimizer written down in closed form via the inverse transformation.
  • In the limit $\lambda\to 1$ the bounds become simple inequalities involving the Shannon entropy and the Kullback-Leibler divergence, and when $\beta=\lambda$ the two inequalities multiply to a Cramér-Rao-like bound in the relative setting.
  • The relative Fisher-Shannon complexity measure defined in (4.20) is monotone under Gaussian convolution (Proposition 4.1), extending a classical monotonicity property to this relative framework.
  • Fixing a desired minimizer $f_*$ and solving for the reference density produces adapted measures whose two factors are each minimized by the reference density $h_*$ while the product of the factors is minimized exactly at $f_*$, a separation of roles not seen in the classical inequalities.
  • Because both new measures are invariant under simultaneous scaling of $f$ and $h$, the inequalities remain valid under arbitrary changes of units, which the previous relative Fisher measures (4.5) and (4.6) did not enjoy.

Reading between the lines

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

  • Beyond the paper: choosing $h$ to be a physically motivated reference density (for example a Gaussian or a known ground-state density) makes the sharp minimizers a parameter-free family of model densities; comparing empirical densities against these minimizers could provide a direct estimator of the nonextensivity parameter $\lambda$.
  • Beyond the paper: the construction is coordinate-based rather than dimension-specific, so the same transformation should produce scaling-invariant relative Fisher and cumulative-moment measures in higher dimensions; verifying the analogues of (4.7)-(4.8) there would be a direct test of the framework.
  • Beyond the paper: the invertibility of $R^{[h]}_1$ connects the sequence of Rényi divergences of $f$ against $h$ to the Rényi entropy powers of the transformed density, suggesting a relative Hausdorff moment problem in which divergence data can recover $f$ by inverting the transformation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper introduces a 'relative differential-escort' transformation R_alpha^[h][f] depending on a reference density h (Definition 3.1), and uses it to define a biparametric relative Fisher divergence F_{p,lambda}[f||h] (Definition 4.1) and relative cumulative moments mu_{p,alpha}[f||h] (Definition 4.2). It proves that these new functionals are scale-invariant (Lemma 4.2) and transfers the classical moment-entropy and triparametric Stam inequalities to the relative setting (Theorem 4.1), with minimizers given by inverse-transformed stretched Gaussians. It also constructs adapted inequalities for a prescribed minimizer by solving for the reference density, leading to explicit formulas involving generalized trigonometric, hyperbolic, and incomplete gamma functions. The main algebraic steps are mostly sound, but the Stam-like inequality as stated has a hypothesis gap in the bounded-support case, and the theorem's parameter domain is not fully consistent with the paper's own definition of Rényi divergence.

Significance. The main conceptual contribution is a scale-invariant relative Fisher measure with sharp inequalities against Rényi and Kullback-Leibler divergences; this is a genuine improvement over earlier relative Fisher information measures (4.5) and (4.6), which scale nontrivially. The algebraic reductions in Lemmas 3.1, 3.2, 4.1 and 4.2 are sound, and the moment-entropy inequality (4.7) follows cleanly from the classical result (2.9). The paper also contains a constructive solution of the inverse minimizer problem, with explicit minimizers in several cases, which is a useful feature. The main caveat is that the Stam-like inequality (4.8) requires additional hypotheses in the bounded-support case, so the central theorem is currently over-stated.

major comments (3)
  1. [§4.2, Theorem 4.1(4.8) and Remark 4.3] The proof applies the classical triparametric Stam inequality (2.7) to F = R_alpha^[h][f]. When 1+beta-lambda > 0, (2.7) is stated for densities that are absolutely continuous on R, not merely on their support. However, F is supported on (0, y_f), and y_f is finite whenever K_alpha[h||f] < infinity (Remark 3.1). For a natural class of admissible pairs with bounded common support and (f/h)(x_f) != 0, the zero extension of F is discontinuous at y_f. For example, with Omega = (0,1), f(x) = 2x, h(x) = 1, alpha = 3/2, one gets y(x) = sqrt(2x) and F(y) = y^3 on (0, sqrt(2)), with F(sqrt(2)) = 2 sqrt(2) > 0; for p = 2, beta = lambda = 2, the sign condition (2.6) holds and 1+beta-lambda = 1 > 0. The condition 'R_alpha^[h][f] absolutely continuous' as stated, and the sufficient conditions in Remark 4.3, neither exclude nor cover this case. Thus (4.8) is not established for such pairs. The authors should either add the requirement that the zero extension is absolutely continuous on R (for example, vanishing endpoint values in the finite-support case), or prove a variant of (2.7) for densities on a bounded interval that accounts for boundary terms and verify that (4.8) holds in the present examples.
  2. [§4.2, Theorem 4.1 (parameter range)] The parameter domain of the inequalities is not consistent with the definition of Rényi divergence. Both (4.7) and (4.8) contain exp(D_xi[f||h]) with xi(lambda, alpha) = 1 + alpha(lambda - 1), but Definition 2.1 defines D_xi only for xi > 0. The hypotheses alpha > 0 and lambda > 1/(1+p*) do not imply xi > 0; for instance, p* = 1, lambda = 0.9, alpha = 20 gives xi = -1. The theorem should either impose alpha(1 - lambda) < 1 when lambda < 1, or explicitly state that the expressions are interpreted via K_xi^{1/(xi-1)} for all real xi for which the integral is finite. As written, part of the stated parameter range is not covered by the paper's own definitions.
  3. [§4.2 and §4.3 (sharpness and minimizers)] The sharpness claim of Theorem 4.1 depends on the finiteness of N_{1/alpha*}[g_{p,lambda}] and N_{1/alpha*}[g_{p,beta,lambda}], which is needed for Definition 3.2, but the paper gives no characterization of this condition and the normalization constants a_{p,lambda} are not given. Consequently the reader cannot determine for which (p, lambda, alpha) the proposed minimizers are admissible, and the explicit formulas in Section 4.3 (for example (4.13)-(4.16)) silently assume this finiteness. A parameter analysis, or at least explicit sufficient conditions for the finiteness of these Rényi entropy powers, should be supplied before the sharpness statement of Theorem 4.1 can be considered complete.
minor comments (4)
  1. [§4.1, Lemma 4.2] The statement writes sigma_{p,lambda} for the relative cumulative moment, while Definition 4.2 defines sigma_{p,alpha}; the subscript should be alpha throughout.
  2. [§4.3] The references to inequalities (4.18) and (4.19) appear before those equations are defined in Section 4.4; they should refer to (4.7) and (4.8).
  3. [§3.1, Lemma 3.2] The notation N_lambda^{1-lambda} in (3.12) is easy to misread; adding a parenthetical or spacing would clarify that it means (N_lambda)^{1-lambda}, as used in the proof and in (4.9).
  4. [§3.2] In the exponential example, the intermediate expression for y(x) contains a cumbersome term of the form a^{alpha-1}(a alpha - a - alpha) that should be simplified or checked; the final support condition alpha < a* is clear, but the derivation would benefit from an extra step.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 4.1 reduces to externally established moment-entropy and Stam inequalities under an invertible change of variables, with no fitted parameters, no self-defining quantities, and no imported uniqueness principle.

full rationale

The derivation chain is self-contained and non-circular. The two claimed inequalities in Theorem 4.1 are obtained by (i) proving direct identities linking the new functionals to classical ones under the relative differential-escort map — Lemma 3.2 (N^{(1−λ)/λ}[f^{[h]}_α] = K_{ξ(λ,α)}[f||h], a change-of-variables computation) and Lemma 4.1 (φ_{p,λ}[f^{[h]}_α] = |α|^{1/λ} φ_{p,λα}[f||h]^α, computed from (3.4)) — and (ii) substituting these identities into the external moment-entropy inequality (2.9) (Lutwak–Yang–Zhang / Bercher) and the triparametric Stam inequality (2.7) applied to the transformed density f^{[h]}_α. Equations (4.7) and (4.8) are literally the classical inequalities restated in the y-coordinate; no target inequality is assumed, and no parameter is fitted to make the constants come out. Sharpness is transferred through the inverse transformation R^{-1,[h]}_α of Definition 3.2, which is invertible when N_{1/α*}[g] < ∞, so equality in the new inequalities corresponds exactly to equality at the stretched Gaussians g_{p,λ} / g_{p,β,λ} in the classical inequalities; this is a legitimate reduction rather than a renaming. The triparametric Stam inequality (2.7) is cited to [42] and [44], and [44] overlaps with two of the present authors, but the cited result is a published, parameter-free theorem with stated hypotheses (sign condition (2.6), absolute continuity) that do not presuppose (4.8); under the scoring rules this self-citation is real independent evidence and does not raise the circularity score. The relative Fisher divergence and relative cumulative moment are defined independently in Definitions 4.1 and 4.2, not in terms of the target divergences, and the adapted inequalities of Section 5 are explicit rewrites of (4.7)/(4.8) for a constructed reference h_⋆. Flagged for the record as a correctness risk rather than a circularity: the proof of (4.8) applies (2.7) to F = R^{[h]}_α[f] when 1+β−λ > 0, a regime where (2.7) demands absolute continuity on R; F has support (0, y_f), finite whenever K_α[h||f] < ∞ (Remark 3.1), and its zero extension generally jumps at y_f (e.g., Ω=(0,1), f(x)=2x, h=1, α=3/2 gives F(y)=y^3 on (0,√2) with F(√2)=2√2>0), a case not covered by the sufficient conditions in Remark 4.3. Likewise, the sharpness claim assumes N_{1/α*}[g_{p,λ}] < ∞ without a parameter analysis. These are hypothesis and completeness gaps in an otherwise honest reduction, not circular steps.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The paper introduces three new mathematical objects: the relative differential-escort transformation and two parametric families of relative measures. No free parameters are fitted; the constants in the inequalities come from known extremals (stretched Gaussians). The main assumptions are standard regularity and support conditions on the densities, plus the validity of the cited classical inequalities.

assumptions (5)
  • domain assumption f and h are probability densities with common support Ω and f,h > 0 on Ω (Eq. 2.1).
    Used throughout; the transformation and inverse rely on positivity and support equality.
  • domain assumption f and h are differentiable on their support for the relative Fisher divergence (Definition 4.1) and R[h]_α[f] is absolutely continuous for the Stam-like inequality (Theorem 4.1 and Remark 4.3).
    Needed for Lemma 3.1 and for applying the classical Stam inequality to the transformed density.
  • standard math The classical moment-entropy inequality (2.9) of Lutwak-Bercher and the triparametric Stam inequality (2.7) from [42,44] are valid under the stated parameter conditions (2.6) and (2.8).
    These are the external benchmarks from which the new inequalities are derived; no proof is given here, only citation.
  • domain assumption The inverse relative differential-escort transformation is well-defined when N_{1/α*}[g] < ∞ (Definition 3.2); for α=1 it requires compact support of g.
    Ensures the inverse transformation with scaling parameter r=N_{1/α*}[g] is well-defined and yields a probability density.
  • ad hoc to paper Definitions 3.1, 4.1 and 4.2 of the transformation and new functionals are adopted.
    The central new constructs; their properties are derived, but their choice is the paper's modeling decision.
invented entities (3)
  • relative differential-escort transformation R[h]_α independent evidence
    purpose: generalizes the differential-escort transformation to depend on a reference density h; basis for all new functionals
    Definition 3.1; its identities (3.12) and (4.2) are derived and can be checked on examples (Section 3.2).
  • relative Fisher divergence F_{p,λ}[f||h] independent evidence
    purpose: scale-invariant biparametric relative Fisher measure that admits sharp bounds by KL/Rényi divergences
    Definition 4.1; at (p,λ)=(2,1) it reproduces known relative Fisher information (4.5), providing an external anchor.
  • relative cumulative moments μ_{p,α}[f||h] independent evidence
    purpose: relative version of cumulative moments, used in the moment-entropy-like inequality
    Definition 4.2; it reduces to the cumulative moments of [40] in the uniform reference case and satisfies scaling invariance (Lemma 4.2).

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Pith. "Pith review of Sharp informational inequalities involving Kullback-Leibler and R\'enyi divergences and a family of scaling-invariant relative Fisher measures." pith.science (2026). https://pith.science/paper/5OX347PV

@misc{pith2026250717408,
  author       = {Pith},
  title        = {Pith review of: Sharp informational inequalities involving Kullback-Leibler and R\'enyi divergences and a family of scaling-invariant relative Fisher measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5OX347PV}},
  note         = {Machine review of arXiv:2507.17408}
}
read the original abstract

We introduce a new transformation called \emph{relative differential-escort}, which extends the usual differential-escort transformation by relating the change of variable to a reference probability density. As an application of it, we define a biparametric family of \emph{relative Fisher measures} presenting significant advantages with respect to the pre-existing ones in the literature: invariance under scaling changes and, consequently, sharp inequalities between the new relative Fisher measure and the well established Kullback-Leibler and R\'enyi divergences. We also introduce a biparametric family of \emph{relative cumulative moment-like measures} and we establish sharp lower bounds of these new measures by the Kullback-Leibler and R\'enyi divergences. The optimal bound and the minimizing densities are given. We also construct a family of inequalities for an arbitrary and fixed minimizing density in which the so-called generalized trigonometric functions plays a central role, providing thus one more interesting application of the newly introduced inequalities and measures.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. New sharp inequalities involving non-relative, relative and cross informational functionals with some remarkable minimizers of generalized Gaussian and Beta types

    cs.IT 2026-07 conditional novelty 4.0 of 10

    Three sharp informational inequalities are derived by combining a Rényi entropy–divergence–cross-entropy inequality with relative Stam and moment-entropy bounds, yielding explicit minimizers of generalized Gaussian an...

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