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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.13423 |
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| _version_ | 1866908348979347456 |
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| author | Cook, William |
| author_facet | Cook, William |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13423 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mixed Fractional Information: Consistency of Dissipation Measures for Stable Laws Cook, William Other Computer Science 60E07, 94A17, 60G51, 26D10, 62B10 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. |
| title | Mixed Fractional Information: Consistency of Dissipation Measures for Stable Laws |
| topic | Other Computer Science 60E07, 94A17, 60G51, 26D10, 62B10 |
| url | https://arxiv.org/abs/2504.13423 |