A definition of the coupled-product for multivariate coupled-exponentials

TL;DR

This paper introduces a multivariate coupled-product and coupled-exponential framework within nonextensive statistical mechanics, enabling the construction of multivariate coupled-Gaussian distributions and modeling distribution coupling.

Contribution

It defines the multivariate coupled-product and coupled-exponential, extending nonextensive calculus to multivariate functions and enabling new distribution constructions.

Findings

Defined multivariate coupled-product with additive dimensions

Constructed multivariate coupled-Gaussian from univariate cases

Generalized the product of independent Gaussians to coupled distributions

Abstract

The coupled-product and coupled-exponential of the generalized calculus of nonextensive statistical mechanics are defined for multivariate functions. The nonlinear statistical coupling is indexed such that k_d = k/(1+dk), where d is the dimensions of the argument of the multivariate coupled-exponential. The coupled-Gaussian distribution is defined such that the argument of the coupled-exponential depends on the coupled-moments but not the coupling parameter. The multivariate version of the coupled-product is defined such that the output dimensions are the sum of the input dimensions. This enables construction of the multivariate coupled-Gaussian from univariate coupled-Gaussians. The resulting construction forms a model of coupling between distributions, generalizing the product of independent Gaussians.