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Through and beyond moments, entropies and Fisher information measures: new informational functionals and inequalities

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Pith's one-line read The paper introduces upper moments and down-Fisher measures—informational functionals built from the up/down transforms—and claims they satisfy sharp inequalities with explicit constants and minimizers, extending moment-entropy, Stam, and…

desk verdict New functionals and inequality statements built by substituting known inequalities through up/down transformations; sound in the classical range, but sharpness and regularity claims outrun what is proved, and the load-bearing identities sit in an unreviewed companion preprint. read the letter →

arxiv 2505.21015 v1 pith:OGH6QKKO submitted 2025-05-27 math-ph cs.ITmath.ITmath.MP

Razvan Gabriel Iagar , David Puertas-Centeno This is my paper
classification math-phcs.ITmath.ITmath.MP MSC 94A1760E1526D1562B10
keywords uppermomentsdown-Fishermeasuresup/downtransformationsinformationalinequalitiesRényientropyFisherinformationmoment-entropyinequalityHausdorffmomentproblem
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 introduces two new families of informational functionals, upper moments and down-Fisher measures, formed by applying classical functionals such as p-moments and Fisher information to densities transformed by the 'up' and 'down' operators introduced in the authors' companion work. It claims that, for these new functionals, the classical moment-entropy, Stam, and Cramér-Rao inequalities admit sharp extensions with explicit optimal constants and minimizers, including in parameter ranges where the underlying density has heavy tails or diverges at the edge of its support. The upshot would be quantitative upper bounds on moment-entropy, Stam, and Cramér-Rao products for a substantially wider class of densities than the stretched Gaussians that minimize the classical versions. The paper also shows that generalized Beta densities play the extremal role for upper moments with fixed moment, and that maximizing upper moments of higher order under fixed lower-order upper-moments extends the MaxEnt approach to the Hausdorff moment problem.

What carries the argument

The machinery is the mutually inverse pair of transformations, up $U_\alpha$ and down $D_\alpha$, taken from the authors' preceding work. The down transform $D_\alpha[f](s) = f^\alpha(x(s))|f'(x(s))|^{-1}$ rewrites a decreasing density in terms of its own derivative, while the up transform is its inverse; together they map heavy-tailed densities to better-behaved ones and vice versa. Lemma 2.1 supplies the load-bearing identities expressing moments, Rényi entropy power, and Fisher information of transformed densities in terms of the original density's functionals. Each new theorem is obtained by substituting these identities into the classical moment-entropy, Stam, and Cramér-Rao inequalities, or into the companion tri-parametric Stam inequality, and then identifying the optimal constant with the classical constant and the minimizer with the transformed classical minimizer.

What would settle it

Take a decreasing heavy-tailed density such as $f(x)=C(1+|x|)^{-\eta}$ with $1<\eta<2$, choose parameters in the mirrored range (4.5), and compute both sides of inequality (4.3) directly; if $(m_{p^*,\alpha}[f]/\sigma_q[f])^{\Theta_1(p,\alpha,q)}$ ever falls below the claimed constant $\kappa^{(-1)}_{p,\alpha,q}$, then the transformation identity fails on that range and the theorem's claim collapses.

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Extended reading notes

Core claim

The central claim is that the up and down transformations act as a dictionary: every classical informational inequality applied to a transformed density becomes a new sharp inequality for the original density, with constants inherited from the classical result and minimizers obtained by pulling the classical minimizer back through the transformation. Concretely, Theorem 4.1 states that for $p$, $q$, $\alpha$ in the classical range (4.4) or the mirrored range (4.5), $(m_{p^*,\alpha}[f]/\sigma_q[f])^{\Theta_1(p,\alpha,q)} \ge \kappa^{(-1)}_{p,\alpha,q}$, with optimal constant and minimizer $D_\alpha[g_{p,\lambda}]$, the down-transform of a stretched Gaussian; Theorems 4.3, 4.4, and 4.6 give analogous sharp upper-moment–entropy, down-Fisher–Fisher, and modified Stam inequalities. The paper argues that these inequalities reveal the same structural relationship between upper moments and moments as exists between moments and Rényi entropy power, and between entropy power and generalized Fisher information, with the down-Fisher measure completing the chain.

Load-bearing premise

The load-bearing premise is that the identities connecting the up and down transforms to moments, entropies, and Fisher information continue to hold on the extended parameter ranges and for heavy-tailed or edge-divergent densities, and the paper takes those identities from its companion work without fully proving the required regularity conditions.

Editorial extensions

If this is right

  • The moment-entropy, Stam, and Cramér-Rao products become bounded above by explicit functions for densities with heavy tails or divergences at the edge of their support, not only for the classical stretched-Gaussian minimizers.
  • The upper-moment–moment inequality (4.3) is saturated by the down-transformed stretched Gaussian $D_\alpha[g_{p,\lambda}]$, a generalized Beta density, placing generalized Beta laws in the role that stretched Gaussians play for ordinary moments.
  • When $\Theta_1(p,\alpha,q^*)>0$, inequality (4.3) yields sharp bounds of the form $\sigma_{q^*}[f] \le A\, m_{p^*,\alpha}[f]$, which in turn give upper bounds for classical moment-entropy and Cramér-Rao products through (4.13) and (4.14).
  • The down-Fisher–Fisher inequality (4.22) and its consequences (4.30)–(4.32) produce upper bounds for Cramér-Rao and Stam products in terms of the down-Fisher measure, including cases with negative parameters in the mirrored domain.
  • Maximizing Rényi entropy under fixed moments is equivalent, via the down transformation, to maximizing a certain upper-moment (or logarithmic-moment) functional under fixed upper-moments, extending the MaxEnt principle to a wider class of reconstruction problems.

Reading between the lines

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

  • Because each proof is a substitution of the transformation identities into a classical inequality, the same recipe would produce additional sharp inequalities for any future inequality whose minimizer is a stretched Gaussian; this generalization is not stated in the paper.
  • The authors' conjecture that only decreasing exponential and power densities survive infinitely many down transforms could be probed numerically by seeking other decreasing densities whose down transform remains decreasing and integrable, which would enlarge the class where the higher-order upper-moment inequalities (4.15) are sharp.
  • A natural extension to higher dimensions or manifolds is left implicit; if the up/down dictionary admits a multidimensional analogue, the same construction would define new complexity measures suitable for quantum position-momentum uncertainty products.
  • Testing the sharpness of the constants on a family of generalized Beta densities with varying edge exponents would offer a concrete numerical check of the optimality claims without needing to verify the full regularity conditions.
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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

5 major / 5 minor

Summary. The paper introduces two new families of informational functionals: (p,α)-upper moments M_{p,α}[f] and their deviations m_{p,α}[f], including higher-order versions, and (p,q,λ)-down-Fisher measures φ_{p,q,λ}[f]. These are obtained by applying classical moments, entropy powers, and Fisher-type information measures to the up/down transformed densities of the companion preprint [38]. The central contribution is a set of informational inequalities: Theorem 4.1 (upper-moment–moment), Theorem 4.2 (iterated upper-moment inequalities), Theorem 4.3 (upper-moment–entropy), Theorem 4.4 (down-Fisher–Fisher), Theorem 4.6 (modified Stam-like inequalities), plus corollaries bounding classical moment–entropy, Stam, and Cramér–Rao products. The proofs are substitution arguments: known moment-entropy, Stam, and Cramér–Rao inequalities are applied to up/down transformed densities, and the transformation identities of Lemma 2.1 are used to rewrite the resulting terms. Applications to the Hausdorff moment problem and MaxEnt-type reconstruction are also discussed.

Significance. If the results are correct in the stated range, the paper gives sharp quantitative inequalities with explicit constants and minimizers for a broad family of densities, including heavy-tailed and boundary-divergent ones, and it connects these to generalized Beta minimizers. The substitution method is transparent and parameter-free: no constants are fitted and no new inequality is assumed, so the derivations are reproducible from the cited classical results and identities. The appendix providing explicit minimizer formulas is a useful service. The main caveats are that the load-bearing transformation identities are imported from a companion preprint without proof, the regularity hypotheses for several theorems are either too weak or left as open problems, and some advertised optimality is explicitly conjectural.

major comments (5)
  1. [Section 2.2, Lemma 2.1] The identities (2.28)–(2.29), together with the α=2 variants, are used in every proof of Section 4 (for example in Eq. (4.12), Theorem 4.3, and Theorem 4.4), but they are stated without proof and without explicit hypotheses beyond 'let f be a probability density'. Since the paper advertises validity for negative p, sub-1 exponents, and densities that diverge at the edge of a compact support, the exact regularity and parameter ranges under which these identities hold must be stated; otherwise the new inequalities inherit every failure of the imported identities. The proof in [38, Section 3.1] should either be reproduced in an appendix or the companion preprint should be made available with the precise hypotheses verified for the extended ranges.
  2. [Theorem 4.1, mirrored range (4.5)] The theorem is stated for 'any probability density function', but in the mirrored case the proof applies the mirrored moment-entropy inequality (2.12), which in Section 2 is stated only for continuously differentiable densities. The up-transformed density f↑_α need not be continuously differentiable for an arbitrary f, so the theorem as stated lacks the regularity hypothesis needed for the application of (2.12). This affects the validity of the result in the mirrored range (4.5), one of the paper's advertised extensions.
  3. [Section 4.1, after Theorem 4.2] The abstract and introduction announce 'optimal constants and minimizers' for the new inequalities, but the paragraph after Theorem 4.2 explicitly states that the iterated minimizer D_{α1}[D_{α0}[g_{p,λ}]] is well defined only under the restriction α0 > 2 - λ/p* - 1/p, and that sharpness in the remaining cases is only conjectured. Thus Theorem 4.2 and the iterated part of Theorem 4.3 (for α1 < 2) are not fully proved as sharp inequalities. The statements and abstract should either be restricted to the proved cases or clearly mark the optimality as conjectural.
  4. [Theorem 4.4] The statement says 'for any probability density function f such that D_{2-λ}[f] is absolutely continuous', but the down transformation is only defined for decreasing densities (Definition 2.1), and for p<1 the tri-parametric Stam inequality (2.16) with condition (2.17) requires f to be continuously differentiable with f' < 0. The theorem omits the monotonicity and differentiability conditions on f, so condition (4.21) alone does not guarantee that D_{2-λ}[f] is well defined. The statement should include the hypotheses that are actually needed for the imported Stam inequality.
  5. [Theorem 4.3, open problem] The open problem stated immediately after Theorem 4.3 acknowledges that the class of densities f for which U_{α0}[f] is absolutely continuous (or of bounded variation) is not characterized. Since Theorem 4.3 is presented as applying 'for any probability density f' under a condition that is left uncharacterized, the practical scope of the theorem is unclear; at minimum, the statement should separate the algebraic validity of the inequality from the currently open question of which densities satisfy the required regularity of U_{α0}[f].
minor comments (5)
  1. [Abstract and Introduction] There is a typographical error in the abstract and repeated in the introduction: 'some of the the most important informational inequalities' should read 'some of the most important informational inequalities'.
  2. [Eq. (2.13)] The notation \widetilde{g}_{p,λ} is defined twice with the same two cases (λ>0 and λ<0); this is harmless but could be simplified to a single definition.
  3. [Section 4.1, Theorem 4.2 proof] The phrase 'we raise the previous inequality to the power α1 − 2' is potentially confusing because when α1 − 2 < 0 the inequality direction changes; the subsequent sentence does address this, but rephrasing would improve clarity.
  4. [Theorem 4.6 proof] There is a stray punctuation artifact in the sentence defining p after Eq. (4.45): '2 + \tilde{p} − β, .' should be cleaned up.
  5. [Appendix] The displayed formulas for g⋆_{p,α,q} in cases (a)–(d) mix the parameters q and λ; aligning the notation explicitly with Eq. (4.2) would help readers check the minimizer expressions.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the new inequalities are transported from classical inequalities by explicit substitutions; self-citation to the companion preprint is load-bearing but non-circular.

full rationale

I find no step in which a claimed new result is assumed as an input or in which a fitted parameter is renamed as a prediction. The paper's constructions are transparent: upper-moments are defined as p-th moments of up-transformed densities (Remark 3.1), and down-Fisher measures are defined so that Lemma 3.1 identifies them with the Fisher information of a down-transformed density. Every new inequality is then obtained by applying a classical inequality (moment-entropy, Cramer-Rao, or Stam) to a transformed density and using the transformation identities of Lemma 2.1. For example, Eq. (4.12) rewrites N_lambda[f^up_alpha] via Lemma 2.1, so Theorem 4.1 is a direct consequence of the moment-entropy inequality (2.10)/(2.12), not a circular restatement of it. The same pattern holds for Theorems 4.3, 4.4, and 4.6: constants and minimizers are transported from the underlying classical inequalities, and no empirical fitting occurs. The paper does rely heavily on the authors' own companion preprint [38] for Lemma 2.1 and for the tri-parametric Stam inequality [38, Theorem 5.1], but those cited statements are parameter-free, have stated assumptions, and do not include the present theorems as conclusions; the dependency is a normal and explicit mathematical reliance on prior work, not a reduction of the present results to their own assumptions. The paper also discloses limitations, including open regularity conditions after Theorem 4.3 and a counterexample to iterated down transformations after Theorem 4.2; these affect completeness and rigor, but they are not circularity. The two-point score reflects the substantial self-citational dependence rather than any detected circular reduction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

No free parameters are fitted to data: the paper is a pure mathematics contribution, and every inequality holds over its stated parameter range rather than being tuned to a dataset. The axioms are the prior informational inequalities, the up/down transformation identities inherited from the authors' companion preprint, and technical regularity conditions for iterated transforms. The new functionals are explicit definitions, not speculative entities, and they come with mathematically checkable predictions (the theorems).

assumptions (5)
  • standard math Classical informational inequalities: bi-parametric Stam (Eq. (2.6)), moment-entropy (Eq. (2.10)), mirrored moment-entropy (Eq. (2.12)), generalized Cramér-Rao (Eq. (2.14)), and tri-parametric Stam (Eq. (2.16)) with their minimizers g_{p,lambda}.
    Taken from [27, 28, 44, 45] and [38, Theorem 5.1]. The proofs of Theorems 4.1, 4.3, 4.4, 4.6 apply these inequalities to transformed densities; the tri-parametric Stam inequality originates in the authors' companion preprint.
  • domain assumption Up/down transformation calculus: Lemma 2.1 identities (Eqs. (2.28) to (2.32)), inversion (Prop. 2.1), scaling (Prop. 2.2), and the double down-transform entropy formula (Prop. 2.3).
    Proven in [38, Section 3.1] and reproduced here without proof. These identities are the engine of the paper: Eq. (4.12) and the proofs of Theorems 4.1, 4.3, 4.4, and 4.6 substitute them verbatim.
  • domain assumption Applicability of the down transformation: f must be strictly decreasing, and for iterated transforms one needs sup_x [f f''/(f')^2] < alpha (Remark 2.1, Proposition 2.3, Theorem 4.6).
    Several statements phrased 'for any probability density' in Theorems 4.2, 4.5, and 4.6 silently carry these regularity constraints. The paper itself notes a counterexample (D_1 on an exponential density) where a second down transformation is impossible, and states an open problem about characterizing the valid class.
  • standard math The extended Fisher functional (Eqs. (2.4) and (2.5)) is well defined for negative and sub-1 exponents p and lambda in the mirrored parameter domain.
    The paper extends the classical (p, lambda)-Fisher information to exponents outside the original definition and uses it in Lemma 2.1 and Section 4. This extension is flagged by the authors as only valid 'when the corresponding functionals are well defined'.
  • standard math Monotonicity of the Rényi entropy power: N_lambda[f] >= N_beta[f] when lambda < beta (Eq. (3.16)).
    Used in the proof of Theorem 3.1 as the starting inequality and referenced to [3]. It is a standard fact for one-dimensional densities.
invented entities (2)
  • (p, alpha)-upper moments and upper deviations M_{p,alpha}[f] and m_{p,alpha}[f], plus higher-order versions M_{p,alpha-vector}[f] independent evidence
    purpose: New functionals defined as ordinary p-moments of the up-transformed density; used to extend moment-type inequalities to heavy-tailed or boundary-divergent densities.
    Definition 3.1/3.3 with identity (3.4). The theorems (e.g., Theorem 4.1) give concrete checkable inequalities for every density in the stated classes, which is a falsifiable handle independent of the definitions.
  • (p, q, lambda)-down-Fisher measures phi_{p,q,lambda}[f] independent evidence
    purpose: Fisher information of the down-transformed density; yields upper bounds on classical Stam, Cramér-Rao, and related informational products.
    Definition 3.4 with identity (3.10). Inequalities in Theorems 3.1, 4.4, and 4.5 provide checkable handles, though the printed statement of Theorem 3.1 appears inconsistent with its proof.

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Cite this review

Pith. "Pith review of Through and beyond moments, entropies and Fisher information measures: new informational functionals and inequalities." pith.science (2026). https://pith.science/paper/OGH6QKKO

@misc{pith2026250521015,
  author       = {Pith},
  title        = {Pith review of: Through and beyond moments, entropies and Fisher information measures: new informational functionals and inequalities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OGH6QKKO}},
  note         = {Machine review of arXiv:2505.21015}
}
abstract

We introduce new classes of informational functionals, called \emph{upper moments}, respectively \emph{down-Fisher measures}, obtained by applying classical functionals such as $p$-moments and the Fisher information to the recently introduced up or down transformed probability density functions. We extend some of the the most important informational inequalities to our new functionals and establish optimal constants and minimizers for them. In particular, we highlight that, under certain constraints, the generalized Beta probability density maximizes (or minimizes) the upper-moments when the moment is fixed. Moreover, we apply these structured inequalities to systematically establish new and sharp upper bounds for the main classical informational products such as moment-entropy, Stam, or Cram\'er-Rao like products under certain regularity conditions. Other relevant properties, such as regularity under scaling changes or monotonicity with respect to the parameter, are studied. Applications to related problems to the Hausdorff moment problem are also given.

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

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