## The Annals of Statistics

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

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

Giovanni Pistone and Carlo Sempi

#### 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