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The Fisher score on the closed simplex

Giovanni Pistone, Fabio Rapallo, Eva Riccomagno

TL;DR

This work extends Fisher score and related information-geometric tools from the open to the closed probability simplex, enabling smooth one-parameter exponential models to hit boundary points where zeros occur. It achieves this by developing an algebraic-statistics framework centered on a fibre bundle of $p$-contrasts and a statistical bundle that encodes velocity via the score, and by introducing exponential transports and natural gradients on the boundary. Key contributions include a boundary-compatible generalization of the Fisher score, a reduced statistical-bundle construction, and a coherent theory of transports and geodesics that preserve or properly modify support. The resulting formalism provides computable, algebraic representations for gradient flows, variational methods, and geometric reasoning in contingency tables and compositional data with zeros, with potential applications to dynamic models and physics-inspired statistical methods.

Abstract

We extend classical analytic tools for finite-state statistical models to allow zero probabilities. Using methods from algebraic statistics and information geometry, we develop a framework in which a smooth statistical model could hit the boundary of the simplex, for example, in contingency tables with non-structural zeros. The central object of our approach is the vector bundle whose fibres are the $p$-contrasts associated to each probability distribution $p$. In this framework, Fisher score and other key statistical concepts, such as entropy for one-dimensional statistical models, admit an algebraic representation also on the boundary of the simplex.

The Fisher score on the closed simplex

TL;DR

This work extends Fisher score and related information-geometric tools from the open to the closed probability simplex, enabling smooth one-parameter exponential models to hit boundary points where zeros occur. It achieves this by developing an algebraic-statistics framework centered on a fibre bundle of -contrasts and a statistical bundle that encodes velocity via the score, and by introducing exponential transports and natural gradients on the boundary. Key contributions include a boundary-compatible generalization of the Fisher score, a reduced statistical-bundle construction, and a coherent theory of transports and geodesics that preserve or properly modify support. The resulting formalism provides computable, algebraic representations for gradient flows, variational methods, and geometric reasoning in contingency tables and compositional data with zeros, with potential applications to dynamic models and physics-inspired statistical methods.

Abstract

We extend classical analytic tools for finite-state statistical models to allow zero probabilities. Using methods from algebraic statistics and information geometry, we develop a framework in which a smooth statistical model could hit the boundary of the simplex, for example, in contingency tables with non-structural zeros. The central object of our approach is the vector bundle whose fibres are the -contrasts associated to each probability distribution . In this framework, Fisher score and other key statistical concepts, such as entropy for one-dimensional statistical models, admit an algebraic representation also on the boundary of the simplex.
Paper Structure (12 sections, 7 theorems, 102 equations, 2 figures)

This paper contains 12 sections, 7 theorems, 102 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Tangent bundle and contrast spaces
  3. Coordinate subspaces
  4. Fisher score
  5. The equation of velocity
  6. Examples
  7. Statistical bundle
  8. Natural gradient
  9. Examples
  10. Geodesics
  11. The score as a velocity in the moving frame
  12. Discussion

Key Result

Theorem 2.1

Let $\mathop{\mathrm{Supp}}\nolimits{v}$ be the support of a vector $v\in \Omega$, namely the subset of $\Omega$ for which $v$ has non zero component. Then

Figures (2)

  • Figure 1: \ref{['ex:entropy-2']}. Entropy with a defective support. Left pane: heat map of the entropy. Right pane: entropy production.
  • Figure 2: Example \ref{['ex:gibbs']} with $U=(0,0,1)$ and $V=(0,1,1.8)$. At $\beta=0$ the curve hits the border point $(1/2,1/2,0)$ with a contact of exponential order. For $\beta\to\pm\infty$, the curve tends to the vertex associated with the maximum and the minimum of $V$, respectively.

Theorems & Definitions (36)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • proof
  • Example 2.1
  • Example 2.2: Example \ref{['ex:quadrato']} continued
  • Example 2.3
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.1
  • ...and 26 more