The Exponential Family in Abstract Information Theory

Naudts, Jan; Anthonis, Ben

doi:10.1007/978-3-642-40020-9_28

Cited by 6 publications

(4 citation statements)

References 14 publications

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“…Phi-divergence and U-divergence coincide with U = exp φ , as noted in [14]. Entropy As mentioned earlier, in the present gauge the rho-tau entropy coincides with the phi-deformed entropy (12).…”

Section: Rho-id Gauge ( = Id = Log ã )supporting

confidence: 81%

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Rho–tau embedding and gauge freedom in information geometry

Naudts

Zhang

2018

Info. Geo.

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The standard model of information geometry, expressed as Fisher-Rao metric and Amari-Chensov tensor, reflects an embedding of probability density by log-transform. The present paper studies parametrized statistical models and the induced geometry using arbitrary embedding functions, comparing single-function approaches (Eguchi's U-embedding and Naudts' deformed-log or phi-embedding) and a twofunction embedding approach (Zhang's conjugate rho-tau embedding). In terms of geometry, the rho-tau embedding of a parametric statistical model defines both a Riemannian metric, called "rho-tau metric", and an alpha-family of rho-tau connections, with the former controlled by a single function and the latter by both embedding functions ρ and τ in general. We identify conditions under which the rho-tau metric becomes Hessian and hence the ±1 rho-tau connections are dually flat. For any choice of rho and tau there exist models belonging to the phi-deformed exponential family for which the rho-tau metric is Hessian. In other cases the rho-tau metric may be only conformally equivalent with a Hessian metric. Finally, we show a formulation of the maximum entropy framework which yields the phi-exponential family as the solution.

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Section: Rho-id Gauge ( = Id = Log ã )supporting

confidence: 81%

“…In fact, it was noted in [14] that the U-divergence and the phi-divergence of the previous section map onto each other when the derivative U of U is considered as a deformed exponential function.…”

Section: U-embedding U Entropy and U Cross-entropymentioning

confidence: 96%

Rho–tau embedding and gauge freedom in information geometry

Naudts

Zhang

2018

Info. Geo.

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“…Indeed, Naudts’ work established deep and fruitful connections between Statistical Physics and Information Geometry [ 8 , 9 , 10 , 11 ]. For instance, both Rényi’s and Tsallis’ entropies are described by Naudts in terms of statistical divergences in the family of q -exponential distributions that includes q -Gaussian distributions, defined in details by Plastino and Vignat [ 11 , 12 , 13 , 14 , 15 ]. The analytic and geometric features of deformed exponentials suggest that they are well suited to model non-normally distributed returns of contingent claims.…”

Section: Introductionmentioning

confidence: 99%

Principal Curves for Statistical Divergences and an Application to Finance

Rodrigues

Cavalcante

2018

Entropy

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This paper proposes a method for the beta pricing model under the consideration of non-Gaussian returns by means of a generalization of the mean-variance model and the use of principal curves to define a divergence model for the optimization of the pricing model. We rely on the q-exponential model so consider the properties of the divergences which are used to describe the statistical model and fully characterize the behavior of the assets. We derive the minimum divergence portfolio, which generalizes the Markowitz's (mean-divergence) approach and relying on the information geometrical aspects of the distributions the Capital Asset Pricing Model (CAPM) is then derived under the geometrical characterization of the distributions which model the data, all by the consideration of principal curves approach. We discuss the possibility of integration of our model into an adaptive procedure that can be used for the search of optimum points on finance applications.

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“…It cannot be negative and vanishes if and only if p = q. In our recent works [2,3,4] we have stressed the importance of considering the statistical manifold M as embedded in the set of all probability distributions. In particular, the divergence function D(p||q), with p not belonging to M, can be used to characterize exponential families.…”

Section: Introductionmentioning

confidence: 99%

Extension of Information Geometry to Non-statistical Systems: Some Examples

Naudts

Anthonis

2015

Lecture Notes in Computer Science

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Our goal is to extend information geometry to situations where statistical modeling is not obvious. The setting is that of modeling experimental data. Quite often the data are not of a statistical nature. Sometimes also the model is not a statistical manifold. An example of the former is the description of the Bose gas in the grand canonical ensemble. An example of the latter is the modeling of quantum systems with density matrices. Conditional expectations in the quantum context are reviewed. The border problem is discussed: through conditioning the model point shifts to the border of the differentiable manifold.

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The Exponential Family in Abstract Information Theory

Cited by 6 publications

References 14 publications

Rho–tau embedding and gauge freedom in information geometry

Rho–tau embedding and gauge freedom in information geometry

Principal Curves for Statistical Divergences and an Application to Finance

Extension of Information Geometry to Non-statistical Systems: Some Examples

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