arXiv subjects

Patrick Michl

Publications and source records attributed to Patrick Michl.

4 recordsLinked to original sources

arXiv2020

A Generalization of the Pearson Correlation to Riemannian Manifolds

The increasing application of deep-learning is accompanied by a shift towards highly non-linear statistical models. In terms of their geometry it is natural to identify these models with Riemannian manifolds. The further analysis of the statistical models therefore raises the issue of a correlation measure, that in the cutting planes of the tangent spaces equals the respective Pearson correlation and extends to a correlation measure that is normalized with respect to the underlying manifold. In this purpose the article reconstitutes elementary properties of the Pearson correlation to successively derive a linear generalization to multiple dimensions and thereupon a nonlinear generalization to principal manifolds, given by the Riemann-Pearson Correlation.

math.ST↗

arXiv2020

Applications of Structural Statistics: Geometric Inference in Exponential Families

Exponential families comprise a broad class of statistical models and parametric families like normal distributions, binomial distributions, gamma distributions or exponential distributions. Thereby the formal representation of its probability distributions induces a confined intrinsic structure, which appears to be that of a dually flat statistical manifold. Conversely it can be shown, that any dually flat statistical manifold, which is given by a regular Bregman divergence uniquely induced a regular exponential family, such that exponential families may - with some restrictions - be regarded as a universal representation of dually flat statistical manifolds. This article reviews the pioneering work of Shun'ichi Amari about the intrinsic structure of exponential families in terms of structural stratistics.

math.ST↗

arXiv2020

Foundations of Structural Statistics: Statistical Manifolds

Upon a consistent topological statistical theory the application of structural statistics requires a quantification of the proximity structure of model spaces. An important tool to study these structures are Pseudo-Riemannian metrices, which in the category of statistical models are induced by statistical divergences. The present article extends the notation of topological statistical models by a differential structure to statistical manifolds and introduces the differential geometric foundations to study distribution families by their differential-, Riemannian- and symplectic geometry.

math.ST↗

arXiv2019

Foundations of Structural Statistics: Topological Statistical Theory

Topological statistical theory provides the foundation for a modern mathematical reformulation of classical statistical theory: Structural Statistics emphasizes the structural assumptions that accompany distribution families and the set of structure preserving transformations between them, given by their statistical morphisms. The resulting language is designed to integrate complicated structured model spaces like deep-learning models and to close the gap to topology and differential geometry. To preserve the compatibility to classical statistics the language comprises corresponding concepts for standard information criteria like sufficiency and completeness.

math.ST↗