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Mixed Fractional Information: Consistency of Dissipation Measures for Stable Laws

William Cook

TL;DR

This work addresses the gap in information theory for symmetric $\alpha$-stable ($S\alphaS$) laws, where classical Fisher information diverges for $\alpha<2$. It defines Mixed Fractional Information (MFI) as the initial rate of relative entropy dissipation along Johnson's stable interpolation between $g_v^{(\alpha)}$ and $g_s^{(\alpha)}$, and proves two equivalent formulations: a chain-rule form based on $D'(v)$ and an integral form involving the MMSE-related score $u(x,0)$ and Fisher-score differences. The central contribution is a rigorous consistency identity $D'(v)=\frac{1}{\alpha v}\mathbb{E}_{g_v}[X(p^F_v-p^F_s)]$, confirming the internal coherence of MFI for $S\alphaS$ inputs; the authors also establish nonnegativity, a closed-form for the Cauchy case ($\alpha=1$), and numerical validation across $\alpha$. By tying entropy dissipation to score-function differences and estimation concepts, the work lays a foundation for a fractional I-MMSE paradigm and for designing functional inequalities tailored to heavy-tailed regimes.

Abstract

Symmetric alpha-stable (S alpha S) distributions with alpha<2 lack finite classical Fisher information. Building on Johnson's framework, we define Mixed Fractional Information (MFI) via the initial rate of relative entropy dissipation during interpolation between S alpha S laws with differing scales, v and s. We demonstrate two equivalent formulations for MFI in this specific S alpha S-to-S alpha S setting. The first involves the derivative D'(v) of the relative entropy between the two S alpha S densities. The second uses an integral expectation E_gv[u(x,0) (pF_v(x) - pF_s(x))] involving the difference between Fisher scores (pF_v, pF_s) and a specific MMSE-related score function u(x,0) derived from the interpolation dynamics. Our central contribution is a rigorous proof of the consistency identity: D'(v) = (1/(alpha v)) E_gv[X (pF_v(X) - pF_s(X))]. This identity mathematically validates the equivalence of the two MFI formulations for S alpha S inputs, establishing MFI's internal coherence and directly linking entropy dissipation rates to score function differences. We further establish MFI's non-negativity (zero if and only if v=s), derive its closed-form expression for the Cauchy case (alpha=1), and numerically validate the consistency identity. MFI provides a finite, coherent, and computable information-theoretic measure for comparing S alpha S distributions where classical Fisher information fails, connecting entropy dynamics to score functions and estimation concepts. This work lays a foundation for exploring potential fractional I-MMSE relations and new functional inequalities tailored to heavy-tailed systems.

Mixed Fractional Information: Consistency of Dissipation Measures for Stable Laws

TL;DR

This work addresses the gap in information theory for symmetric -stable () laws, where classical Fisher information diverges for . It defines Mixed Fractional Information (MFI) as the initial rate of relative entropy dissipation along Johnson's stable interpolation between and , and proves two equivalent formulations: a chain-rule form based on and an integral form involving the MMSE-related score and Fisher-score differences. The central contribution is a rigorous consistency identity , confirming the internal coherence of MFI for inputs; the authors also establish nonnegativity, a closed-form for the Cauchy case (), and numerical validation across . By tying entropy dissipation to score-function differences and estimation concepts, the work lays a foundation for a fractional I-MMSE paradigm and for designing functional inequalities tailored to heavy-tailed regimes.

Abstract

Symmetric alpha-stable (S alpha S) distributions with alpha<2 lack finite classical Fisher information. Building on Johnson's framework, we define Mixed Fractional Information (MFI) via the initial rate of relative entropy dissipation during interpolation between S alpha S laws with differing scales, v and s. We demonstrate two equivalent formulations for MFI in this specific S alpha S-to-S alpha S setting. The first involves the derivative D'(v) of the relative entropy between the two S alpha S densities. The second uses an integral expectation E_gv[u(x,0) (pF_v(x) - pF_s(x))] involving the difference between Fisher scores (pF_v, pF_s) and a specific MMSE-related score function u(x,0) derived from the interpolation dynamics. Our central contribution is a rigorous proof of the consistency identity: D'(v) = (1/(alpha v)) E_gv[X (pF_v(X) - pF_s(X))]. This identity mathematically validates the equivalence of the two MFI formulations for S alpha S inputs, establishing MFI's internal coherence and directly linking entropy dissipation rates to score function differences. We further establish MFI's non-negativity (zero if and only if v=s), derive its closed-form expression for the Cauchy case (alpha=1), and numerically validate the consistency identity. MFI provides a finite, coherent, and computable information-theoretic measure for comparing S alpha S distributions where classical Fisher information fails, connecting entropy dynamics to score functions and estimation concepts. This work lays a foundation for exploring potential fractional I-MMSE relations and new functional inequalities tailored to heavy-tailed systems.

Paper Structure

This paper contains 33 sections, 3 theorems, 55 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Motivation: Extending Information Theory to Stable Laws
  3. Symmetric alpha-Stable Distributions
  4. Symmetry Assumption.
  5. Parameter Range.
  6. Relative score notation.
  7. Johnson’s Fractional de Bruijn Identity Framework
  8. Related Work and Positioning
  9. Contributions
  10. Mixed Fractional Information and Score Function
  11. Definition of MFI and Regularity Conditions
  12. Connection to Integral Form
  13. Analytical Score Function for SalphaS Initial Density
  14. Properties of MFI for S-alpha-S Densities
  15. Chain Rule Formulation
  16. ...and 18 more sections

Key Result

Lemma 2.3

For the interpolation starting at $h_0 = g_v^{(\alpha)}$, the score function $u(x, t)$ from Eq. eq:pde at time $t$ is: and the initial score function is:

Figures (1)

  • Figure 1: Structure of the MFI integral for symmetric $\alpha$-stable densities under scale interpolation, with $\alpha = 1.5$, initial scale $v = 1.2$, and target scale $s = 1.0$. Top: the MMSE-related score $u(x, 0) = \frac{x (v - s)}{s v}$. Middle: the difference of Fisher scores $\Delta\mathrm{pF}(x) = \mathrm{pF}_{v}(x) - \mathrm{pF}_{s}(x)$. Bottom: the integrand $g_v(x)\, u(x, 0)\, \Delta\mathrm{pF}(x)$ whose integral over $\mathbb{R}$ gives the MFI.

Theorems & Definitions (13)

  • Definition 2.1: Regularity Class H_alpha
  • Definition 2.2: Mixed Fractional Information
  • Lemma 2.3
  • proof
  • Remark 2.4: On “conditional expectation” for $\alpha\le1$
  • Remark 2.5: Specificity of the Linear Score
  • Proposition 3.1: Consistency Identity
  • proof
  • Remark 3.2: Scope of the Consistency Identity
  • Theorem 3.3
  • ...and 3 more