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Moment polytopes of toric exponential families

Mathieu Molitor

TL;DR

This work identifies the moment polytope of the torification $N$ of an exponential family on a finite sample space as an affine projection of a simplex. By combining Dombrowski's Kähler structure on the tangent bundle, parallel lattices, and a lifting to a holomorphic isometric immersion into complex projective space, the authors show $J(N)=-4\pi T(\Delta_{m})+C$ for some integer matrix $T$ and constant $C$ when the exponential family is full. The approach bridges information geometry (marginal/polytope descriptions) with toric/Kähler geometry, including explicit connections to the Veronese embedding in the binomial case. The results provide a concrete geometric description of moment polytopes for toric exponential families, framing them as projections of high-dimensional simplices and enabling explicit computations via affine maps.

Abstract

We show that the moment polytope of a Kähler toric manifold, constructed as the torification (in the sense of M. Molitor, Kähler toric manifolds from dually flat spaces, arXiv:2109.04839, 2021) of an exponential family defined on a finite sample space, is the projection of a higher-dimensional simplex.

Moment polytopes of toric exponential families

TL;DR

This work identifies the moment polytope of the torification of an exponential family on a finite sample space as an affine projection of a simplex. By combining Dombrowski's Kähler structure on the tangent bundle, parallel lattices, and a lifting to a holomorphic isometric immersion into complex projective space, the authors show for some integer matrix and constant when the exponential family is full. The approach bridges information geometry (marginal/polytope descriptions) with toric/Kähler geometry, including explicit connections to the Veronese embedding in the binomial case. The results provide a concrete geometric description of moment polytopes for toric exponential families, framing them as projections of high-dimensional simplices and enabling explicit computations via affine maps.

Abstract

We show that the moment polytope of a Kähler toric manifold, constructed as the torification (in the sense of M. Molitor, Kähler toric manifolds from dually flat spaces, arXiv:2109.04839, 2021) of an exponential family defined on a finite sample space, is the projection of a higher-dimensional simplex.

Paper Structure

This paper contains 8 sections, 7 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Torification of dually flat manifolds
  3. Dombrowski's construction.
  4. Parallel lattices.
  5. Torification.
  6. Proof of Theorem \ref{['nfekwnkefnk']}
  7. Momentum map and marginal polytope.
  8. Kähler immersion into $\mathbb{P}^{m}$.

Key Result

Theorem 1.1

Suppose $\mathcal{E}$ is full and $\textup{J} :N\to \mathbb{R}^{n}$ is closed (meaning that the image of every closed set is closed). Then there exist an $n\times m$ matrix $T$ with integer entries and $C\in \mathbb{R}^{n}$ such that where $\Delta_{m}$ is the $m$-simplex given by $\{(x_{1},...,x_{m})\in \mathbb{R}^{m}\,\,\vert\,\, x_{k}\geq 0\,\, \textup{for all $k$ and}\,x_{1}+...+x_{m}\leq 1\}

Theorems & Definitions (13)

  • Theorem 1.1
  • Corollary 1.2
  • Example 1.3
  • Definition 2.1: Torification
  • Theorem 2.2: Equivalence of regular torifications
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • ...and 3 more