# One-dimensional exponential families with constant Hessian scalar curvature

## TL;DR

This paper classifies one-dimensional exponential families with finite sample spaces that have constant Hessian scalar curvature, revealing a specific relationship between the curvature and integer parameters, and highlighting the binomial distribution's role.

## Contribution

It provides a complete classification of 1D exponential families with constant Hessian scalar curvature, identifying the curvature values and the significance of the binomial distribution.

## Key findings

- Hessian scalar curvature is quantized as 2/k for positive integers k
- Binomial distribution plays a central role in the classification
- Explicit characterization of exponential families with constant curvature

## Abstract

We give a complete classification of 1-dimensional exponential families $\mathcal{E}$ defined over a finite space $\Omega=\{x_{0}, ...,x_{n}\}$ whose Hessian scalar curvature is constant. We observe an interesting phenomenon: if $\mathcal{E}$ has constant Hessian scalar curvature, say $\lambda$, then $\lambda=\tfrac{2}{k}$ for some positive integer $k\leq m$. We also discuss the central role played by the binomial distribution in this classification.