The Annals of Statistics

An Infinite-Dimensional Geometric Structure on the Space of all the Probability Measures Equivalent to a Given One

Giovanni Pistone and Carlo Sempi

Full-text: Open access

Abstract

Let $\mathscr{M}_\mu$ be the set of all probability densities equivalent to a given reference probability measure $\mu$. This set is thought of as the maximal regular (i.e., with strictly positive densities) $\mu$-dominated statistical model. For each $f \in \mathscr{M}_\mu$ we define (1) a Banach space $L_f$ with unit ball $\mathscr{V}_f$ and (2) a mapping $s_f$ from a subset $\mathscr{U}_f$ of $\mathscr{M}_\mu$ onto $\mathscr{V}_f$, in such a way that the system $(s_f, \mathscr{U}_f, f \in \mathscr{M}_\mu)$ is an affine atlas on $\mathscr{M}_\mu$. Moreover each parametric exponential model dominated by $\mu$ is a finite-dimensional affine submanifold and each parametric statistical model dominated by $\mu$ with a suitable regularity is a submanifold. The global geometric framework given by the manifold structure adds some insight to the so-called geometric theory of statistical models. In particular, the present paper gives some of the developments connected with the Fisher information metrics (Rao) and the Hilbert bundle introduced by Amari.

Article information

Source
Ann. Statist., Volume 23, Number 5 (1995), 1543-1561.

Dates
First available in Project Euclid: 11 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176324311

Digital Object Identifier
doi:10.1214/aos/1176324311

Mathematical Reviews number (MathSciNet)
MR1370295

Zentralblatt MATH identifier
0848.62003

JSTOR
links.jstor.org

Subjects
Primary: 62A25

Keywords
Nonparametric statistical manifolds Orlicz spaces

Citation

Pistone, Giovanni; Sempi, Carlo. An Infinite-Dimensional Geometric Structure on the Space of all the Probability Measures Equivalent to a Given One. Ann. Statist. 23 (1995), no. 5, 1543--1561. doi:10.1214/aos/1176324311. https://projecteuclid.org/euclid.aos/1176324311


Export citation