# Statistical Einstein manifolds of exponential families with group-invariant potential functions

## TL;DR

This paper classifies statistical Einstein manifolds within exponential families by deriving PDEs for potential functions and finding solutions using symmetry methods, advancing understanding in information geometry.

## Contribution

It introduces a classification framework for statistical Einstein manifolds in exponential families using PDEs and Lie symmetry techniques.

## Key findings

- Derived PDEs for potential functions of exponential families.
- Obtained group-invariant solutions via Lie symmetry reductions.
- Provided new insights into the geometric structure of statistical models.

## Abstract

This paper mainly contributes to a classification of statistical Einstein manifolds, namely statistical manifolds at the same time are Einstein manifolds. A statistical manifold is a Riemannian manifold, each of whose points is a probability distribution. With the Fisher information metric as a Riemannian metric, information geometry was developed to understand the intrinsic properties of statistical models, which play important roles in statistical inference, etc. Among all these models, exponential families is one of the most important kinds, whose geometric structures are fully determined by their potential functions. To classify statistical Einstein manifolds, we derive partial differential equations for potential functions of exponential families; special solutions of these equations are obtained through the ansatz method as well as group-invariant solutions via reductions using Lie point symmetries.