Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Divergence and Deformed Exponential Family

Hiroshi Matsuzoe, Asuka Takatsu

TL;DR

The paper generalizes information-geometric foundations by introducing the $(h,\tau)$-divergence and $(h,\tau)$-exponential families. It establishes a sufficient condition under which the induced metric and flat connection form a Hessian structure, and it frames a corresponding canonical divergence and Pythagorean relation. It then develops the theory of identically repeatable deformed exponential families and proves a law of large numbers for $(h,\tau)$-dependent sequences, including explicit constructions and probabilistic bounds. These results broaden the scope of information geometry to deformed entropies and nonstandard divergences, enabling generalized statistical modeling and convergence analyses beyond the classical KL/exponential-family setting.

Abstract

The Kullback--Leibler divergence together with exponential families establishes the foundation of information geometry and is widely generalized. Among the generalization, we focus on the $(h,τ)$-divergence and $(h,τ)$-exponential families. We present a sufficient condition for the $(h,τ)$-divergence to induce a Hessian structure on an $(h,τ)$-exponential family. We also define the $(h,τ)$-dependence of random variables and prove a kind of the law of large numbers.

Divergence and Deformed Exponential Family

TL;DR

The paper generalizes information-geometric foundations by introducing the -divergence and -exponential families. It establishes a sufficient condition under which the induced metric and flat connection form a Hessian structure, and it frames a corresponding canonical divergence and Pythagorean relation. It then develops the theory of identically repeatable deformed exponential families and proves a law of large numbers for -dependent sequences, including explicit constructions and probabilistic bounds. These results broaden the scope of information geometry to deformed entropies and nonstandard divergences, enabling generalized statistical modeling and convergence analyses beyond the classical KL/exponential-family setting.

Abstract

The Kullback--Leibler divergence together with exponential families establishes the foundation of information geometry and is widely generalized. Among the generalization, we focus on the -divergence and -exponential families. We present a sufficient condition for the -divergence to induce a Hessian structure on an -exponential family. We also define the -dependence of random variables and prove a kind of the law of large numbers.
Paper Structure (11 sections, 23 theorems, 174 equations)

This paper contains 11 sections, 23 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Statistical model
  5. Hessian structure
  6. Contrast function
  7. $(h,\tau)$-divergence
  8. Deformed Exponential Family
  9. Proof of Theorem \ref{['mainthm2']}
  10. Pythagorean relation
  11. Proof of Theorem \ref{['mainthm3']}

Key Result

Theorem 1.3

Let $(h, \tau, I)\in \mathcal{G}$ and $\mathcal{M}$ be an $(h,\tau)$-exponential family in $\mathcal{P}_I(\mu)$. If $\mathcal{X}$ is compact and $\mathbb{I}_\tau$ is constant on $\mathcal{M}$, then $( g^{h,\tau},\nabla^{h,\tau})$ is a Hessian structure on $\mathcal{M}$ and its global potential is $-

Theorems & Definitions (66)

  • Definition 1.1
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.7
  • ...and 56 more