Exponential statistical manifold

Cena, Alberto; Pistone, Giovanni

doi:10.1007/s10463-006-0096-y

Cited by 73 publications

(105 citation statements)

References 11 publications

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“…We refer to [10] and [23,Chapter 2] for more details on the matter. We consider the Young function Φ : R x → Φ(x) = cosh x − 1 and, for any p ∈ P > , the Orlicz space L Φ (p) = L cosh −1 (p) is defined as follows: a real random variable…”

Section: Model Spacesmentioning

confidence: 99%

“…Definition 2 (Statistical exponential manifold [10,Definition 20]). For p ∈ P > , the statistical exponential manifold at p is…”

Section: Exponential Manifoldmentioning

confidence: 99%

“…The basic case of a finite state space has been extended by Amari and coworkers to the case of a parametric set of strictly positive probability densities on a generic sample space. Following a suggestion by Dawid in [7,8], a particular nonparametric version of that theory was developed in a series of papers [9][10][11][12][13][14][15][16][17][18][19], where the set P > of all strictly positive probability densities of a measure space is shown to be a Banach manifold (as it is defined in [20][21][22]) modeled on an Orlicz Banach space, see, e.g., [23, Chapter II].…”

Section: Introductionmentioning

confidence: 99%

“…This material is included for convenience only and this part should be skipped by any reader aware of any of the papers [9][10][11][12][13][14][15][16][17][18][19] quoted above. The following Section 3 is mostly based on the same references and it is intended to introduce that manifold structure and to give a first example of application to the study of Kullback-Liebler divergence.…”

Section: Introductionmentioning

confidence: 99%

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Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation

Lods

Pistone

2015

Entropy

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Information Geometry generalizes to infinite dimension by modeling the tangent space of the relevant manifold of probability densities with exponential Orlicz spaces. We review here several properties of the exponential manifold on a suitable set E of mutually absolutely continuous densities. We study in particular the fine properties of the Kullback-Liebler divergence in this context. We also show that this setting is well-suited for the study of the spatially homogeneous Boltzmann equation if E is a set of positive densities with finite relative entropy with respect to the Maxwell density. More precisely, we analyze the Boltzmann operator in the geometric setting from the point of its Maxwell's weak form as a composition of elementary operations in the exponential manifold, namely tensor product, conditioning, marginalization and we prove in a geometric way the basic facts, i.e., the H-theorem. We also illustrate the robustness of our method by discussing, besides the Kullback-Leibler divergence, also the property of Hyvärinen divergence. This requires us to generalize our approach to Orlicz-Sobolev spaces to include derivatives.

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Section: Model Spacesmentioning

confidence: 99%

“…Definition 2 (Statistical exponential manifold [10,Definition 20]). For p ∈ P > , the statistical exponential manifold at p is…”

Section: Exponential Manifoldmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation

Lods

Pistone

2015

Entropy

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“…As an analogue of part (i) of Theorem 1.1, we can construct a symplectic structure on the square of maximal exponential manifolds. We also obtain a local isomorphism from Diff k (S 1 )/S 1 , the space of L 2 k probability densities on the [3] Symplectic structures on statistical manifolds 373 unit circle S 1 , to the cotangent bundle of a Lagrangian submanifold of Diff k (S 1 )/S 1 . This is an analogue of part (ii) of Theorem 1.1 in the nonparametric case.…”

Section: Introductionmentioning

confidence: 99%

Symplectic Structures on Statistical Manifolds

Noda

2011

J. Aust. Math. Soc.

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A relationship between symplectic geometry and information geometry is studied. The square of a dually flat space admits a natural symplectic structure that is the pullback of the canonical symplectic structure on the cotangent bundle of the dually flat space via the canonical divergence. With respect to the symplectic structure, there exists a moment map whose image is the dually flat space. As an example, we obtain a duality relation between the Fubini-Study metric on a projective space and the Fisher metric on a statistical model on a finite set. Conversely, a dually flat space admitting a symplectic structure is locally symplectically isomorphic to the cotangent bundle with the canonical symplectic structure of some dually flat space. We also discuss nonparametric cases.2010 Mathematics subject classification: primary 53D20; secondary 60D05.

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Geometric mean of probability measures and geodesics of Fisher information metric

Itoh

Satoh

2023

Mathematische Nachrichten

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The space of all probability measures having positive density function on a connected compact smooth manifold 𝑀, denoted by (𝑀), carries the Fisher information metric 𝐺. We define the geometric mean of probability measures by the aid of which we investigate information geometry of (𝑀), equipped with 𝐺. We show that a geodesic segment joining arbitrary probability measures 𝜇 1 and 𝜇 2 is expressed by using the normalized geometric mean of its endpoints. As an application, we show that any two points of (𝑀) can be joined by a unique geodesic. Moreover, we prove that the function 𝓁 defined by 𝓁 ( 𝜇 1 , 𝜇 2 ) ∶= 2 arccos ∫ 𝑀 √ 𝑝 1 𝑝 2 𝑑𝜆, 𝜇 𝑖 = 𝑝 𝑖 𝜆, 𝑖 = 1, 2, gives the Riemannian distance function on (𝑀). It is shown that geodesics are all minimal.

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Exponential statistical manifold

Abstract: Information geometry, Statistical manifold, Orlicz space, Moment generating functional, Cumulant generating functional, Kullback–Leibler divergence,

Cited by 73 publications

References 11 publications

Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation

Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation

Symplectic Structures on Statistical Manifolds

Geometric mean of probability measures and geodesics of Fisher information metric

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