Mixed Fractional Information: Consistency of Dissipation Measures for Stable Laws

2025-04-18

AI Summary

Classical Fisher information is infinite for symmetric α-stable (SαS) distributions with α < 2 due to their heavy tails, rendering it incapable of quantifying scale differences. Method: This paper introduces the Mixed Fractional Information (MFI), a finite, computable information measure defined as the initial rate of relative entropy dissipation under scale interpolation. Contribution/Results: We rigorously prove a consistency identity for MFI, establishing its intrinsic connections to fractional score differences and an MMSE-type scoring function. We derive two equivalent analytical expressions for MFI—demonstrating its non-negativity and proving that it vanishes if and only if the scales are equal—and obtain a closed-form solution for the Cauchy case (α = 1). Numerical experiments confirm MFI’s robustness and computational feasibility. As the first self-consistent fractional information framework for heavy-tailed systems, MFI enables the development of fractional I-MMSE relations and novel functional inequalities.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Content-based information diffusion
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Defines Mixed Fractional Information for stable laws
• Proves consistency between entropy dissipation formulations
• Provides computable measure for heavy-tailed distributions

Innovation

• Defines Mixed Fractional Information via entropy dissipation
• Proves consistency identity for MFI equivalence
• Provides finite measure for S alpha S distributions