## Bernoulli

• Bernoulli
• Volume 5, Number 4 (1999), 721-760.

### The exponential statistical manifold: mean parameters, orthogonality and space transformations

#### Abstract

Let $( X,cal X,μ)$ be a measure space, and let $cal M (X,cal X,μ)$ denote the set of the $μ$-almost surely strictly positive probability densities. It was shown by Pistone and Sempi in 1995 that the global geometry on $cal M (X,cal X,μ)$ can be realized by an affine atlas whose charts are defined locally by the mappings $cal M (X,cal X,μ)⊃cal U p∋q↦log(q/p)+K(p,q)∈B p$, where $cal U p$ is a suitable open set containing $p$, $K (p,q)$ is the Kullback--Leibler relative information and $B p$ is the vector space of centred and exponentially $( p⋅μ)$-integrable random variables. In the present paper we study the transformation of such an atlas and the related manifold structure under basic transformations, i.e. measurable transformation of the sample space. A generalization of the mixed parametrization method for exponential models is also presented.

#### Article information

Source
Bernoulli, Volume 5, Number 4 (1999), 721-760.

Dates
First available in Project Euclid: 19 February 2007

https://projecteuclid.org/euclid.bj/1171899326

Mathematical Reviews number (MathSciNet)
MR1704564

Zentralblatt MATH identifier
0947.62003

#### Citation

Pistone, Giovanni; Piera Rogantin, Maria. The exponential statistical manifold: mean parameters, orthogonality and space transformations. Bernoulli 5 (1999), no. 4, 721--760. https://projecteuclid.org/euclid.bj/1171899326

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