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Generalized informational functionals and new monotone measures of statistical complexity

Razvan Gabriel Iagar, David Puertas-Centeno

TL;DR

The paper develops a biparametric framework of informational transformations that extends up/down and differential-escort mappings, introducing down-moments and cumulative upper-moments as new informational functionals. It establishes sharp inequalities, including mirrored-domain variants, and identifies minimizers frequently expressible through generalized trigonometric functions, linking moments, Rényi entropies, and Fisher information. It then defines several measures of statistical complexity based on these functionals and proves monotonicity properties under algebraic conjugations of the transformations, revealing an underlying group structure. The results unify classical and mirrored parameter regimes, offering interpolation between fundamental informational quantities and providing a robust toolset for studying informational inequalities and complexity in continuous distributions with potential applications in information theory and related fields.

Abstract

In this paper we introduce a biparametric family of transformations which can be seen as an extension of the so-called up and down transformations. This new class of transformations allows to us to introduce new informational functionals, which we have called \textit{down-moments} and \textit{cumulative upper-moments}. A remarkable fact is that the down-moments provide, in some cases, an interpolation between the $p$-th moments and the power Rényi entropies of a probability density. We establish new and sharp inequalities relating these new functionals to the classical informational measures such as moments, Rényi and Shannon entropies and Fisher information measures. We also give the optimal bounds as well as the minimizing densities, which are in some cases expressed in terms of the generalized trigonometric functions. We furthermore define new classes of measures of statistical complexity obtained as quotients of the new functionals, and establish monotonicity properties for them through an algebraic conjugation of up and down transformations. All of these properties highlight an intricate structure of functional inequalities.

Generalized informational functionals and new monotone measures of statistical complexity

TL;DR

The paper develops a biparametric framework of informational transformations that extends up/down and differential-escort mappings, introducing down-moments and cumulative upper-moments as new informational functionals. It establishes sharp inequalities, including mirrored-domain variants, and identifies minimizers frequently expressible through generalized trigonometric functions, linking moments, Rényi entropies, and Fisher information. It then defines several measures of statistical complexity based on these functionals and proves monotonicity properties under algebraic conjugations of the transformations, revealing an underlying group structure. The results unify classical and mirrored parameter regimes, offering interpolation between fundamental informational quantities and providing a robust toolset for studying informational inequalities and complexity in continuous distributions with potential applications in information theory and related fields.

Abstract

In this paper we introduce a biparametric family of transformations which can be seen as an extension of the so-called up and down transformations. This new class of transformations allows to us to introduce new informational functionals, which we have called \textit{down-moments} and \textit{cumulative upper-moments}. A remarkable fact is that the down-moments provide, in some cases, an interpolation between the -th moments and the power Rényi entropies of a probability density. We establish new and sharp inequalities relating these new functionals to the classical informational measures such as moments, Rényi and Shannon entropies and Fisher information measures. We also give the optimal bounds as well as the minimizing densities, which are in some cases expressed in terms of the generalized trigonometric functions. We furthermore define new classes of measures of statistical complexity obtained as quotients of the new functionals, and establish monotonicity properties for them through an algebraic conjugation of up and down transformations. All of these properties highlight an intricate structure of functional inequalities.

Paper Structure

This paper contains 19 sections, 18 theorems, 224 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary tools
  3. Basic informational measures
  4. Generalized trigonometric functions
  5. Differential-escort transformation and cumulative moments
  6. Up and down transformations, upper-moments and down-Fisher measures
  7. Previous informational inequalities
  8. Biparametric up and down transforms: definitions
  9. Biparametric up and down transformations
  10. Cumulative upper-moments
  11. Down-moments
  12. Informational inequalities
  13. Cumulative moment-entropy inequality in the mirrored domain
  14. A down-moment--Fisher inequality
  15. A generalized cumulative moment-entropy inequality
  16. ...and 4 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.

Figures (1)

  • Figure 1: Graphical scheme of the connections between classical and mirrored domains.

Theorems & Definitions (42)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • Lemma 2.1
  • Definition 2.3: Upper-moments
  • Definition 2.4: Down-Fisher measures
  • Definition 3.1
  • Definition 3.2
  • Remark 3.1
  • Proposition 3.1
  • ...and 32 more