The Fisher score on the closed simplex

TL;DR

This paper extends classical Fisher score analysis to models on the boundary of the probability simplex, incorporating algebraic and geometric methods to handle zero probabilities in contingency tables.

Contribution

It introduces a novel algebraic and geometric framework for Fisher score and entropy on the boundary of the simplex, accommodating zero probabilities in statistical models.

Findings

Framework handles zero probabilities in contingency tables.

Fisher score and entropy are represented algebraically on the boundary.

Extends classical tools to models with non-structural zeros.

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.