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

Razvan Gabriel Iagar, David Puertas-Centeno

TL;DR

The paper addresses the limitation of classical informational functionals when handling densities with heavy tails or edge divergences by introducing two new constructs: the $(p,\alpha)$-upper-moments and the down-Fisher measures $\varphi_{p,q,\lambda}$, defined through successive up/down transformations. It develops a rigorous framework of sharp informational inequalities—relating these new functionals to classical ones such as moments, entropy power, and Fisher information—and characterizes optimal constants and extremizers (notably stretched Gaussians and generalized Beta densities). The authors demonstrate a twofold structure of inequalities (one-step and two-step) and derive upper bounds for generalized Stam products, with applications to the Hausdorff moment problem and a generalized MaxEnt principle that leverages upper-moments. Overall, the work broadens the class of densities amenable to information-theoretic analysis and provides a versatile toolkit for reconstruction and uncertainty quantification in broader settings.

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 Cramér-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.

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

TL;DR

The paper addresses the limitation of classical informational functionals when handling densities with heavy tails or edge divergences by introducing two new constructs: the -upper-moments and the down-Fisher measures , defined through successive up/down transformations. It develops a rigorous framework of sharp informational inequalities—relating these new functionals to classical ones such as moments, entropy power, and Fisher information—and characterizes optimal constants and extremizers (notably stretched Gaussians and generalized Beta densities). The authors demonstrate a twofold structure of inequalities (one-step and two-step) and derive upper bounds for generalized Stam products, with applications to the Hausdorff moment problem and a generalized MaxEnt principle that leverages upper-moments. Overall, the work broadens the class of densities amenable to information-theoretic analysis and provides a versatile toolkit for reconstruction and uncertainty quantification in broader settings.

Abstract

We introduce new classes of informational functionals, called \emph{upper moments}, respectively \emph{down-Fisher measures}, obtained by applying classical functionals such as -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 Cramér-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.

Paper Structure

This paper contains 16 sections, 22 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Preliminary notions and results
  3. A brief review of moments and entropies
  4. Up and down transformations
  5. Upper-moments and down-Fisher measures: basic properties
  6. Upper-moments: the first order case
  7. Upper-moments: higher order cases
  8. Down-Fisher measures: basic properties
  9. Order relation between down-Fisher measures
  10. Dealing with the moment problems
  11. Informational inequalities
  12. Upper-moment--moment inequality
  13. Upper-moment--entropy inequality
  14. Down-Fisher--Fisher inequality
  15. Upper bounds for the generalized Stam product
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $f$ a probability density and let ${f}^\downarrow_{\alpha}=\mathfrak {D}_{\alpha}[f]$ and ${f}^\uparrow_{\alpha}=\mathfrak {U}_{\alpha}[f]$ be its $\alpha$-order down and up transformations. Then, up to a translation, that is, $\mathfrak {D}_{\alpha}\mathfrak {U}_{\alpha}=\mathfrak {U}_{\alpha}\mathfrak {D}_{\alpha}=\mathbb I$, where $\mathbb I$ denotes the identity operator (up to a translat

Theorems & Definitions (47)

  • Definition 2.1
  • Remark 2.1: Derivatives and composition
  • Definition 2.2
  • Proposition 2.1
  • Proposition 2.2: Scaling and up/down transformations
  • Lemma 2.1
  • Proposition 2.3
  • proof
  • Definition 3.1
  • Remark 3.1
  • ...and 37 more