Statistical exponential families: A digest with flash cards

TL;DR

This paper provides a concise overview of exponential family distributions, their properties, dualities, and geometric structures, serving as a comprehensive reference for statisticians and researchers.

Contribution

It offers a summarized, organized digest of exponential family distributions, including definitions, properties, decompositions, and geometric perspectives, with plans for ongoing updates.

Findings

Summarizes key properties of exponential families

Details decompositions and formulas of common distributions

Explores geometric structures like Fisher-Rao and dual affine connections

Abstract

This document describes concisely the ubiquitous class of exponential family distributions met in statistics. The first part recalls definitions and summarizes main properties and duality with Bregman divergences (all proofs are skipped). The second part lists decompositions and related formula of common exponential family distributions. We recall the Fisher-Rao-Riemannian geometries and the dual affine connection information geometries of statistical manifolds. It is intended to maintain and update this document and catalog by adding new distribution items.