Entropic Measure on Multidimensional Spaces

TL;DR

This paper introduces an entropic measure on compact manifolds of any dimension, constructed via a conjugation map applied to the Dirichlet process, extending a 1D measure concept to higher dimensions.

Contribution

It defines a new entropic measure on multidimensional spaces using a conjugation map, generalizing a 1D probability measure transformation to higher dimensions.

Findings

Constructed the entropic measure $bP^eta$ on compact manifolds.

Established the conjugation map as a continuous involution.

Provided an heuristic interpretation of the measure as a Gibbs measure.

Abstract

We construct the entropic measure $\mathbb{P}^\beta$ on compact manifolds of any dimension. It is defined as the push forward of the Dirichlet process (another random probability measure, well-known to exist on spaces of any dimension) under the {\em conjugation map} $$\Conj:\mathcal{P}(M)\to\mathcal{P}(M).$$ This conjugation map is a continuous involution. It can be regarded as the canonical extension to higher dimensional spaces of a map between probability measures on 1-dimensional spaces characterized by the fact that the distribution functions of $\mu$ and $\Conj(\mu)$ are inverse to each other. We also present an heuristic interpretation of the entropic measure as $$d\mathbb{P}^\beta(\mu)=\frac{1}{Z}\exp(-\beta\cdot {Ent} (\mu|m))\cdot d\mathbb{P}^0(\mu).$$