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Jaynes-Gibbs Entropic Convex Duals and Orthogonal Polynomials

The univariate noncentral distributions can be derived by multiplying their central distributions with translation factors. When constructed in terms of translated uniform distributions on unit radius hyperspheres, these translation factors become generating functions for classical families of ortho...

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Autor principal: Le Blanc, Richard
Formato: Online Artículo Texto
Lenguaje:English
Publicado: MDPI 2022
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9141810/
https://www.ncbi.nlm.nih.gov/pubmed/35626592
http://dx.doi.org/10.3390/e24050709
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author Le Blanc, Richard
author_facet Le Blanc, Richard
author_sort Le Blanc, Richard
collection PubMed
description The univariate noncentral distributions can be derived by multiplying their central distributions with translation factors. When constructed in terms of translated uniform distributions on unit radius hyperspheres, these translation factors become generating functions for classical families of orthogonal polynomials. The ultraspherical noncentral t, normal N, F, and [Formula: see text] distributions are thus found to be associated with the Gegenbauer, Hermite, Jacobi, and Laguerre polynomial families, respectively, with the corresponding central distributions standing for the polynomial family-defining weights. Obtained through an unconstrained minimization of the Gibbs potential, Jaynes’ maximal entropy priors are formally expressed in terms of the empirical densities’ entropic convex duals. Expanding these duals on orthogonal polynomial bases allows for the expedient determination of the Jaynes–Gibbs priors. Invoking the moment problem and the duality principle, modelization can be reduced to the direct determination of the prior moments in parametric space in terms of the Bayes factor’s orthogonal polynomial expansion coefficients in random variable space. Genomics and geophysics examples are provided.
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spelling pubmed-91418102022-05-28 Jaynes-Gibbs Entropic Convex Duals and Orthogonal Polynomials Le Blanc, Richard Entropy (Basel) Article The univariate noncentral distributions can be derived by multiplying their central distributions with translation factors. When constructed in terms of translated uniform distributions on unit radius hyperspheres, these translation factors become generating functions for classical families of orthogonal polynomials. The ultraspherical noncentral t, normal N, F, and [Formula: see text] distributions are thus found to be associated with the Gegenbauer, Hermite, Jacobi, and Laguerre polynomial families, respectively, with the corresponding central distributions standing for the polynomial family-defining weights. Obtained through an unconstrained minimization of the Gibbs potential, Jaynes’ maximal entropy priors are formally expressed in terms of the empirical densities’ entropic convex duals. Expanding these duals on orthogonal polynomial bases allows for the expedient determination of the Jaynes–Gibbs priors. Invoking the moment problem and the duality principle, modelization can be reduced to the direct determination of the prior moments in parametric space in terms of the Bayes factor’s orthogonal polynomial expansion coefficients in random variable space. Genomics and geophysics examples are provided. MDPI 2022-05-16 /pmc/articles/PMC9141810/ /pubmed/35626592 http://dx.doi.org/10.3390/e24050709 Text en © 2022 by the author. https://creativecommons.org/licenses/by/4.0/Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
spellingShingle Article
Le Blanc, Richard
Jaynes-Gibbs Entropic Convex Duals and Orthogonal Polynomials
title Jaynes-Gibbs Entropic Convex Duals and Orthogonal Polynomials
title_full Jaynes-Gibbs Entropic Convex Duals and Orthogonal Polynomials
title_fullStr Jaynes-Gibbs Entropic Convex Duals and Orthogonal Polynomials
title_full_unstemmed Jaynes-Gibbs Entropic Convex Duals and Orthogonal Polynomials
title_short Jaynes-Gibbs Entropic Convex Duals and Orthogonal Polynomials
title_sort jaynes-gibbs entropic convex duals and orthogonal polynomials
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9141810/
https://www.ncbi.nlm.nih.gov/pubmed/35626592
http://dx.doi.org/10.3390/e24050709
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