Characteristic Functionals of Dirichlet Measures

TL;DR

This paper analyzes the characteristic functionals of Dirichlet measures, revealing their algebraic structures and dependence properties, with implications for understanding Dirichlet processes and their applications.

Contribution

It computes characteristic functionals of Dirichlet-Ferguson measures, identifies their symmetry algebra, and explores the structure of Dirichlet posteriors, extending to infinite-dimensional cases.

Findings

Characteristic functionals of Dirichlet-Ferguson measures are explicitly computed.

The symmetry algebra of the Dirichlet distribution's characteristic functional is identified as a simple Lie algebra of type A.

The lattice of Dirichlet posteriors has a natural Lie algebra module structure.

Abstract

We compute characteristic functionals of Dirichlet-Ferguson measures over a locally compact Polish space and prove continuous dependence of the random measure on the parameter measure. In finite dimension, we identify the dynamical symmetry algebra of the characteristic functional of the Dirichlet distribution with a simple Lie algebra of type $A$. We study the lattice determined by characteristic functionals of categorical Dirichlet posteriors, showing that it has a natural structure of weight Lie algebra module and providing a probabilistic interpretation. A partial generalization to the case of the Dirichlet-Ferguson measure is also obtained.