Authors : John Armstrong, Shun-Ichi Amari
DOI URL : http://dx.doi.org/10.1007/978-3-319-25040-3_27
Video : http://www.youtube.com/watch?v=lgnw-X0MUvE
Slides: Armstrong_Pontryagin forms Hessian.pdf
Presentation : https://www.see.asso.fr/node/14280
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Abstract:
We show that Hessian manifolds of dimensions 4 and above must have vanishing Pontryagin forms. This gives a topological obstruction to the existence of Hessian metrics. We find an additional explicit curvature identity for Hessian 4-manifolds. By contrast, we show that all analytic Riemannian 2-manifolds are Hessian.

Author : Barbara Opozda
DOI URL : http://dx.doi.org/10.1007/978-3-319-25040-3_26
Video : http://www.youtube.com/watch?v=b8TsyqMJJaM
Slides: Opozda_curvature statistical structure.pdf
Presentation : https://www.see.asso.fr/node/14279
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Abstract:
Curvature properties for statistical structures are studied. The study deals with the curvature tensor of statistical connections and their duals as well as the Ricci tensor of the connections, Laplacians and the curvature operator. Two concepts of sectional curvature are introduced. The meaning of the notions is illustrated by presenting few exemplary theorems.

Authors : Charles Cavalcante, David de Souza, Rui Vigelis
DOI URL : http://dx.doi.org/10.1007/978-3-319-25040-3_25
Video : No Autorisation
Slides : No Autorisation
Presentation : https://www.see.asso.fr/node/14278
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Abstract:
We define a metric and a family of α-connections in statistical manifolds, based on ϕ-divergence, which emerges in the framework of ϕ-families of probability distributions. This metric and α-connections generalize the Fisher information metric and Amari’s α-connections. We also investigate the parallel transport associated with the α-connection for α = 1.

Authors : Hiroshi Matsuzoe, Monta Sakamoto
DOI URL : http://dx.doi.org/10.1007/978-3-319-25040-3_79
Video : http://www.youtube.com/watch?v=hGc1z8EYR24
Slides: Matsuzoe-generalization independence Student.pdf
Presentation : https://www.see.asso.fr/node/14272
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Abstract:
In anomalous statistical physics, deformed algebraic structures are important objects. Heavily tailed probability distributions, such as Student’s t-distributions, are characterized by deformed algebras. In addition, deformed algebras cause deformations of expectations and independences of random variables. Hence, a generalization of independence for multivariate Student’s t-distribution is studied in this paper. Even if two random variables which follow to univariate Student’s t-distributions are independent, the joint probability distribution of these two distributions is not a bivariate Student’s t-distribution. It is shown that a bivariate Student’s t-distribution is obtained from two univariate Student’s t-distributions under q-deformed independence.

Authors : Tomonari Sei, Ushio Tanaka
DOI URL : http://dx.doi.org/10.1007/978-3-319-25040-3_78
Video : http://www.youtube.com/watch?v=IpvuWcxHX4M
Slides: Sei_Geometric Properties textile plot.pdf
Presentation : https://www.see.asso.fr/node/14271
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Abstract:
The textile plot proposed by Kumasaka and Shibata (2008) is a method for data visualization. The method transforms a data matrix in order to draw a parallel coordinate plot. In this paper, we investigate a set of matrices induced by the textile plot, which we call the textile set, from a geometrical viewpoint. It is shown that the textile set is written as the union of two differentiable manifolds if data matrices are restricted to be full-rank.

Author : Ali Mohammad-Djafari
DOI URL : http://dx.doi.org/10.1007/978-3-319-25040-3_77
Video : http://www.youtube.com/watch?v=8ePQ4NKTG-A
Slides: Mahammad-Djafari_variational Bayesian approximation.pdf
Presentation : https://www.see.asso.fr/node/14270
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Abstract:
Clustering, classification and Pattern Recognition in a set of data are between the most important tasks in statistical researches and in many applications. In this paper, we propose to use a mixture of Student-t distribution model for the data via a hierarchical graphical model and the Bayesian framework to do these tasks. The main advantages of this model is that the model accounts for the uncertainties of variances and covariances and we can use the Variational Bayesian Approximation (VBA) methods to obtain fast algorithms to be able to handle large data sets.

Authors : Damiano Brigo, John Armstrong
DOI URL : http://dx.doi.org/10.1007/978-3-319-25040-3_76
Video : http://www.youtube.com/watch?v=BPkO4sDSeE4
Slides: Brigo_Stochastic PDE projection.pdf
Presentation : https://www.see.asso.fr/node/14269
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Abstract:
We review the manifold projection method for stochastic nonlinear filtering in a more general setting than in our previous paper in Geometric Science of Information 2013. We still use a Hilbert space structure on a space of probability densities to project the infinite dimensional stochastic partial differential equation for the optimal filter onto a finite dimensional exponential or mixture family, respectively, with two different metrics, the Hellinger distance and the L2 direct metric. This reduces the problem to finite dimensional stochastic differential equations. In this paper we summarize a previous equivalence result between Assumed Density Filters (ADF) and Hellinger/Exponential projection filters, and introduce a new equivalence between Galerkin method based filters and Direct metric/Mixture projection filters. This result allows us to give a rigorous geometric interpretation to ADF and Galerkin filters. We also discuss the different finite-dimensional filters obtained when projecting the stochastic partial differential equation for either the normalized (Kushner-Stratonovich) or a specific unnormalized (Zakai) density of the optimal filter.

Authors : Frank Critchley, Germain Van Bever, Paul Marriott, Radka Sabolova
Video : http://www.youtube.com/watch?v=lpzCTiW1QHI
Sides: https://drive.google.com/open?id=0B0QKxsVtOaTiMjVPTWowM2J5U2M
Presentation : https://www.see.asso.fr/node/14268
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Authors : Osamu Komori, Shinto Eguchi
DOI URL : http://dx.doi.org/10.1007/978-3-319-25040-3_66
Video : http://www.youtube.com/watch?v=KooWAyot6U0
Slides: Komori_path connectedness.pdf
Presentation : https://www.see.asso.fr/node/14267
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Abstract:
We introduce a class of paths or one-parameter models connecting arbitrary two probability density functions (pdf’s). The class is derived by employing the Kolmogorov-Nagumo average between the two pdf’s. There is a variety of such path connectedness on the space of pdf’s since the Kolmogorov-Nagumo average is applicable for any convex and strictly increasing function. The information geometric insight is provided for understanding probabilistic properties for statistical methods associated with the path connectedness. The one-parameter model is extended to a multidimensional model, on which the statistical inference is characterized by sufficient statistics.

Author : Hiroto Inoue
DOI URL : http://dx.doi.org/10.1007/978-3-319-25040-3_65
Video : http://www.youtube.com/watch?v=sWl6blVP7UY
Slides: Inoue_Group Theoretical study.pdf
Presentation : https://www.see.asso.fr/node/14266
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Abstract:
We consider the geodesic equation on the elliptical model, which is a generalization of the normal model. More precisely, we characterize this manifold from the group theoretical view point and formulate Eriksen’s procedure to obtain geodesics on normal model and give an alternative proof for it.