On φ-Families of Probability Distributions

Vigelis, Rui F.; Cavalcante, Charles C.

doi:10.1007/s10959-011-0400-5

Cited by 36 publications

(33 citation statements)

References 11 publications

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“…Here we built our model spaces according to the proposal of Vigelis and Cavalcante [62]. if φ(x) = x.…”

Section: Model Spacementioning

confidence: 99%

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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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“…Here we built our model spaces according to the proposal of Vigelis and Cavalcante [62]. if φ(x) = x.…”

Section: Model Spacementioning

confidence: 99%

“…if φ(x) = x. Note that the φ-exponential notation is not used in [62], where φ denotes a class of deformed exponential function larger than the one used here.…”

Section: Model Spacementioning

confidence: 99%

Nonparametric Information Geometry

Pistone

2013

Lecture Notes in Computer Science

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“…The function ϕ is used there in the definition of the (Musielak-Orlicz) model spaces, and gives rise to dual divergence functions distinct from D α . Here, our aim is somewhat different from those of [16] and [24]. We provide a simple framework for the classical information geometry in infinite dimensions; this requires a stronger topology on the model space than that associated with the ϕ-function ψ.…”

Section: Discussionmentioning

confidence: 99%

“…The implicit function theorem shows that Z : G → R, defined by Z(a) = Ψ −1 a (1), is of class C l . According to (24), its first derivative takes the form:…”

Section: The Manifolds Of Probability Measuresmentioning

confidence: 99%

Infinite-dimensional statistical manifolds based on a balanced chart

Newton

2016

Bernoulli

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We develop a family of infinite-dimensional Banach manifolds of measures on an abstract measurable space, employing charts that are "balanced" between the density and log-density functions. The manifolds, $(\tilde{M}_{\lambda},\lambda\in [2,\infty))$, retain many of the features of finite-dimensional information geometry; in particular, the $\alpha$-divergences are of class $C^{\lceil\lambda\rceil-1}$, enabling the definition of the Fisher metric and $\alpha$-derivatives of particular classes of vector fields. Manifolds of probability measures, $(M_{\lambda},\lambda\in [2,\infty))$, based on centred versions of the charts are shown to be $C^{\lceil\lambda \rceil-1}$-embedded submanifolds of the $\tilde{M}_{\lambda}$. The Fisher metric is a pseudo-Riemannian metric on $\tilde{M}_{\lambda}$. However, when restricted to finite-dimensional embedded submanifolds it becomes a Riemannian metric, allowing the full development of the geometry of $\alpha$-covariant derivatives. $\tilde{M}_{\lambda}$ and $M_{\lambda}$ provide natural settings for the study and comparison of approximations to posterior distributions in problems of Bayesian estimation.Comment: Published at http://dx.doi.org/10.3150/14-BEJ673 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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“…Still in the non-quantum case, the authors in [22] proposed a generalization of the exponential family of probabilities by replacing the ordinary exponential function by a deformed exponential function ϕ(·), which is a function that respects some properties and presents a more flexible model for the probability density functions.…”

Section: Introductionmentioning

confidence: 99%

A generalized quantum relative entropy

Andrade¹,

Vigelis²,

Cavalcante³

2020

AMC

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We propose a generalization of the quantum relative entropy by considering the geodesic on a manifold formed by all the invertible density matrices P. This geodesic is defined from a deformed exponential function ϕ which allows to work with a wider class of families of probability distributions. Such choice allows important flexibility in the statistical model. We show and discuss some properties of this proposed generalized quantum relative entropy.2020 Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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On φ-Families of Probability Distributions

Cited by 36 publications

References 11 publications

Nonparametric Information Geometry

Nonparametric Information Geometry

Infinite-dimensional statistical manifolds based on a balanced chart

A generalized quantum relative entropy

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