Information geometry of the space of probability measures and barycenter maps

In this article, we present recent developments of information geometry, namely, geometry of the Fisher metric, dualistic structures, and divergences on the space of probability measures, particularly the theory of geodesics of the Fisher metric. Moreover, we consider several facts concerning the barycenter of probability measures on the ideal boundary of a Hadamard manifold from a viewpoint of the information geometry.

Holographic units membrane and fisher metric

Holographic units membrane and fisher metric

Conformal action and fisher metric

Conformal action and conformal fisher metric

Coarse-graining of the Einstein–Hilbert action rewritten by the Fisher information metric

In this study, considering the Fisher information metric (Fisher metric) given by a specific form, which is the form of weights in statistics, we rewrite the Einstein–Hilbert (EH) action. Then, determining the transformation rules of the Fisher metric, etc under the coarse-graining, we perform the coarse-graining toward that rewritten EH action. We finally show an existence of a trivial fixed-point. Here, the existence of a trivial fixed-point is not trivial for us because we consider the metric given by the Fisher metric, which is not the normal metric and has to satisfy some constraint in the formalism of the Fisher metric. We use the path-integral in our analysis. At this time we have to accept that a fundamental constraint in the formalism of the Fisher metric is broken at the quantum level. However we consider we can accept this with the thought that some constraints and causal relations held at the classical level usually get broken at the quantum level. We finds some problems of the Fisher metric. The space–time we consider in this study is two-dimensional.

Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics

In this paper, we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau’s symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum information geometry. We emphasize the role of equivariance with respect to Lie group actions and the role of several concepts from geometric mechanics, such as momentum maps, Casimir functions, coadjoint orbits, and Lie-Poisson brackets with cocycles, as unifying structures appearing in various applications of this framework to information geometry and machine learning. For instance, we discuss the expression of the Fisher metric in presence of equivariance and we exploit the property of the entropy of the Souriau model as a Casimir function to apply a geometric model for energy preserving entropy production. We illustrate this framework with several examples including multivariate Gaussian probability densities, and the Bogoliubov-Kubo-Mori metric as a quantum version of the Fisher metric for quantum information on coadjoint orbits. We exploit this geometric setting and Lie group equivariance to present symplectic and multisymplectic variational Lie group integration schemes for some of the equations associated with Souriau symplectic and polysymplectic models, such as the Lie-Poisson equation with cocycle.

Lie Group Cohomology and (Multi)Symplectic Integrators : New Geometric Tools for Lie Group Machine Learning based on Souriau Geometric Statistical Mechanics

In this paper we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau's symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum information geometry. We emphasize the role of equivariance with respect to Lie group actions and the role of several concepts from geometric mechanics, such as momentum maps, Casimir functions, coadjoint orbits, and Lie-Poisson brackets with cocycles, as unifying structures appearing in various applications of this framework to information geometry and machine learning. For instance, we discuss the expression of the Fisher metric in presence of equivariance and we exploit the property of the entropy of the Souriau model as a Casimir function to apply a geometric model for energy preserving entropy production. We illustrate this framework with several examples including multivariate Gaussian probability densities, and the Bogoliubov-Kubo-Mori metric as a quantum version of the Fisher metric for quantum information on coadjoint orbits. We exploit this geometric setting and Lie group equivariance to present symplectic and multisymplectic variational Lie group integration schemes for some of the equations associated to Souriau symplectic and polysymplectic models, such as the Lie-Poisson equation with cocycle.

Diffeological Statistical Models, the Fisher Metric and Probabilistic Mappings

We introduce the notion of a C k -diffeological statistical model, which allows us to apply the theory of diffeological spaces to (possibly singular) statistical models. In particular, we introduce a class of almost 2-integrable C k -diffeological statistical models that encompasses all known statistical models for which the Fisher metric is defined. This class contains a statistical model which does not appear in the Ay–Jost–Lê–Schwachhöfer theory of parametrized measure models. Then, we show that, for any positive integer k , the class of almost 2-integrable C k -diffeological statistical models is preserved under probabilistic mappings. Furthermore, the monotonicity theorem for the Fisher metric also holds for this class. As a consequence, the Fisher metric on an almost 2-integrable C k -diffeological statistical model P ⊂ P ( X ) is preserved under any probabilistic mapping T : X ⇝ Y that is sufficient w.r.t. P. Finally, we extend the Cramér–Rao inequality to the class of 2-integrable C k -diffeological statistical models.

Action for entanglement string

the world surface of a tangled string is the world's information density in the form of a Fisher metric. Such an approach allows the connection of string theory with the idea of the entanglement of the world surface and the entanglement of space-time.