Conformal-Projective Geometry of Statistical Manifolds

Kurose, Takashi

doi:10.4036/iis.2002.89

Cited by 23 publications

(20 citation statements)

References 5 publications

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“…165-215] gave a modern differential geometric treatment of statistical problems by introducing the notion of an abstract statistical manifold. This generalization has been considered in many papers by other geometers ( [6,24,29,30,49,50] and etc.). In this section, we give an overview of the geometry of such manifold.…”

Section: An Abstract Statistical Manifoldmentioning

confidence: 99%

A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

Mikeš

Stepanova

2013

Ann Glob Anal Geom

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In the present paper, we consider a five-dimensional Riemannian manifold with an irreducible SO(3)-structure as an example of an abstract statistical manifold. We prove that if a five-dimensional Riemannian manifold with an irreducible SO(3)-structure is a statistical manifold of constant curvature, then the metric of the Riemannian manifold is an Einstein metric. In addition, we show that a five-dimensional Euclidean sphere with an irreducible SO(3)-structure cannot be a conjugate symmetric statistical manifold. Finally, we show some results for a five-dimensional Riemannian manifold with a nearly integrable SO(3)-structure. For example, we prove that the structure tensor of a nearly integrable SO(3)-structure on a five-dimensional Riemannian manifold is a harmonic symmetric tensor and it defines the first integral of third order of the equations of geodesics. Moreover, we consider some topological properties of five-dimensional compact and conformally flat Riemannian manifolds with irreducible SO(3)-structure.

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Section: An Abstract Statistical Manifoldmentioning

confidence: 99%

A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

Mikeš

Stepanova

2013

Ann Glob Anal Geom

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“…Further, it has a close relation with the α-geometry [8], which is one of information geometric structure of constant curvature. This new dually flat structure, different from the old one given rise to from the invariancy in information geometry, can be also obtained by conformal flattening of the α-geometry [11,12], using a technique in the conformal and projective geometry [13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Geometry of q-Exponential Family of Probability Distributions

Амари

Ohara

2011

Entropy

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The Gibbs distribution of statistical physics is an exponential family of probability distributions, which has a mathematical basis of duality in the form of the Legendre transformation. Recent studies of complex systems have found lots of distributions obeying the power law rather than the standard Gibbs type distributions. The Tsallis q-entropy is a typical example capturing such phenomena. We treat the q-Gibbs distribution or the q-exponential family by generalizing the exponential function to the q-family of power functions, which is useful for studying various complex or non-standard physical phenomena. We give a new mathematical structure to the q-exponential family different from those previously given. It has a dually flat geometrical structure derived from the Legendre transformation and the conformal geometry is useful for understanding it. The q-version of the maximum entropy theorem is naturally induced from the q-Pythagorean theorem. We also show that the maximizer of the q-escort distribution is a Bayesian MAP (Maximum A posteriori Probability) estimator.

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“…We remark that several fundamental results in conformal-projective geometry have been obtained in Kurose's recent work [3]. Tchebychev geometry has interesting applications to Bayesian statistics.…”

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confidence: 99%

“…We can obtain the result from straightforward calculations. See also Section 2 in [6] or Section 2 in [3].…”

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Traceless cubic forms on statistical manifolds and Tchebychev geometry

Matsuzoe

2005

PDEs, Submanifolds and Affine Differential Geometry

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Abstract. Geometry of traceless cubic forms is studied. It is shown that the traceless part of the cubic form on a statistical manifold determines a conformal-projective equivalence class of statistical manifolds. This conformal-projective equivalence on statistical manifolds is a natural generalization of conformal equivalence on Riemannian manifolds. As an application, Tchebychev type immersions in centroaffine immersions of codimension two are studied.1. Introduction. The purpose of this paper is to study geometry of traceless cubic forms on statistical manifolds. Cubic forms and their traceless parts have been studied in affine differential geometry. In particular, it is known that an affine hypersurface lies on a quadratic if its cubic form vanishes everywhere on the hypersurface (e.g. [1]).In affine hypersurface theory, the traceless cubic form is invariant under choice of equiaffine transversal vector fields. (In other words, we often say that a traceless cubic form is invariant under relative normalizations.) We generalize this property, namely, we show that the traceless cubic form on a statistical manifold is invariant under conformalprojective changes of statistical manifolds. Moreover, we show that the traceless cubic form determines a conformal-projective class of statistical manifolds. The notion of conformal-projective class of statistical manifolds was introduced by the present author in [6]. This class induces conformal-projective geometry of statistical manifolds, which is a natural generalization of conformal geometry of Riemannian manifolds.Traceless cubic forms are useful objects in Tchebychev hypersurface theory, and conformal-projective relations are useful objects in geometry of centroaffine immersions

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Conformal-Projective Geometry of Statistical Manifolds

Cited by 23 publications

References 5 publications

A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

Geometry of q-Exponential Family of Probability Distributions

Traceless cubic forms on statistical manifolds and Tchebychev geometry

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