Geometry of extended exponential models

Imparato, Daniele; Trivellato, Barbara

doi:10.1017/cbo9780511642401.027

Cited by 3 publications

(3 citation statements)

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“…We feel that the non parametric approach is of interest even in the case of a finite state space. We build upon our previous work in this field, namely [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. Other contributions are referred to in the text below.…”

Section: Introductionmentioning

confidence: 99%

Nonparametric Information Geometry

Pistone

2013

Lecture Notes in Computer Science

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Abstract. The differential-geometric structure of the set of positive densities on a given measure space has raised the interest of many mathematicians after the discovery by C.R. Rao of the geometric meaning of the Fisher information. Most of the research is focused on parametric statistical models. In series of papers by author and coworkers a particular version of the nonparametric case has been discussed. It consists of a minimalistic structure modeled according the theory of exponential families: given a reference density other densities are represented by the centered log likelihood which is an element of an Orlicz space. This mappings give a system of charts of a Banach manifold. It has been observed that, while the construction is natural, the practical applicability is limited by the technical difficulty to deal with such a class of Banach spaces. It has been suggested recently to replace the exponential function with other functions with similar behavior but polynomial growth at infinity in order to obtain more tractable Banach spaces, e.g. Hilbert spaces. We give first a review of our theory with special emphasis on the specific issues of the infinite dimensional setting. In a second part we discuss two specific topics, differential equations and the metric connection. The position of this line of research with respect to other approaches is briefly discussed.

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Section: Introductionmentioning

confidence: 99%

Nonparametric Information Geometry

Pistone

2013

Lecture Notes in Computer Science

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“…Many statistical models can be seen as exponential sub-models; see, e.g. Imparato and Trivellato (2009). Here we focus only on the conditional expectation model, treated also in Pistone and Rogantin (1999).…”

Section: Density Projections and Exponential Sub-modelsmentioning

confidence: 99%

Exponential models by Orlicz spaces and applications

2018

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“…More recently, different authors have generalized this structure replacing the exponential function with deformed exponentials (see, e.g., Vigelis and Cavalcante [ 2 ], De Andrade et al [ 3 ]) and in other directions (see e.g., Imparato and Trivellato [ 4 ]).…”

Section: Introductionmentioning

confidence: 99%