Jaynes-Gibbs Entropic Convex Duals and Orthogonal Polynomials

Area of Science:

• Statistical theory
• Mathematical physics
• Probability theory

Background:

• Univariate noncentral distributions are typically derived from central distributions using translation factors.
• Orthogonal polynomials, such as Gegenbauer, Hermite, Jacobi, and Laguerre, are fundamental in various mathematical and statistical applications.
• Jaynes' maximal entropy principle provides a framework for constructing probability distributions based on constraints.

Purpose of the Study:

• To establish a formal connection between univariate noncentral distributions and classical orthogonal polynomial families.
• To develop an efficient method for determining Jaynes-Gibbs priors using orthogonal polynomial expansions.
• To demonstrate the applicability of the developed methodology in fields like genomics and geophysics.

Main Methods:

• Derivation of noncentral distributions by multiplying central distributions with translation factors, expressed via translated uniform distributions on hyperspheres.
• Formal expression of Jaynes' maximal entropy priors as entropic convex duals of empirical densities.
• Expansion of entropic convex duals on orthogonal polynomial bases to determine Jaynes-Gibbs priors.
• Application of the moment problem and duality principle for modelization in parametric and random variable spaces.

Main Results:

• The ultraspherical noncentral t, normal N, F, and chi-squared distributions are formally associated with Gegenbauer, Hermite, Jacobi, and Laguerre polynomial families, respectively.
• Translation factors for noncentral distributions act as generating functions for these orthogonal polynomial families.
• Jaynes-Gibbs priors can be efficiently determined by expanding their entropic convex duals on orthogonal polynomial bases.
• Modelization is simplified by directly determining prior moments in parametric space using Bayes factor expansion coefficients.

Conclusions:

• A clear mathematical link exists between univariate noncentral distributions and orthogonal polynomials, unifying statistical and polynomial theories.
• The proposed method offers an expedient and formal approach to constructing Jaynes-Gibbs priors.
• The methodology is broadly applicable, with demonstrated utility in complex domains such as genomics and geophysics.