Geometric modeling in probability and statistics pdf

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Published: 20.06.2021  Documentation Help Center. The geometric distribution is a one-parameter family of curves that models the number of failures before one success in a series of independent trials, where each trial results in either success or failure, and the probability of success in any individual trial is constant. For example, if you toss a coin, the geometric distribution models the number of tails observed before the result is heads.

Probability And Statistics Solved Problems Pdf

This book covers topics of Informational Geometry, a field which deals with the differential geometric study of the manifold probability density functions. This is a field that is increasingly attracting the interest of researchers from many different areas of science, including mathematics, statistics, geometry, computer science, signal processing, physics and neuroscience.

This textbook is a unified presentation of differential geometry and probability theory, and constitutes a text for a course directed at graduate or advanced undergraduate students interested in applications of differential geometry in probability and statistics. The book contains over proposed exercises meant to help students deepen their understanding, and it is accompanied by software that is able to provide numerical computations of several information geometric objects. The reader will understand a flourishing field of mathematics in which very few books have been written so far.

Springer Professional. Back to the search result list. Statistical Models Abstract. This chapter presents the notion of statistical models, a structure associated with a family of probability distributions, which can be given a geometric structure.

This chapter deals with statistical models given parametrically. By specifying the parameters of a distribution, we determine a unique element of the family.

When the family of distributions can be described smoothly by a set of parameters, this can be considered as a multidimensional surface. We are interested in the study of the properties that do not depend on the choice of model coordinates. Some Fisher metrics will involve the use of non-elementary functions, such as the digamma and trigamma functions.

Entropy is a notion taken form Thermodynamics, where it describes the uncertainty in the movement of gas particles. In this chapter the entropy will be considered as a measure of uncertainty of a random variable. Even if the entropy of a finite, discrete density is always positive, in the case of continuous density the entropy is not always positive. This drawback can be corrected by introducing another concept, which measures the relative entropy between two given densities.

This chapter studies the Kullback—Leibler relative entropy known also as the Kullback—Leibler divergence between two probability densities in both discrete and continuous cases. The informational energy is a concept inspired from the kinetic energy expression of Classical Mechanics. From the information theory point of view, the informational energy is a measure of uncertainty or randomness of a probability system, and was introduced and studied for the first time by Onicescu [67, 68] in the mids.

This chapter is dedicated to the study of entropy maximization under moment constraints. We present results of entropy maximization under constraints of mean, variance, or any N moments. The solution of these variational problems belongs to the exponential family.

However, explicit solutions exist only in a few particular cases. A distinguished role is played by the study of the Maxwell—Boltzmann distribution. This chapter contains a brief introduction to the classical theory of differential geometry. The fundamental notions presented here deal with differentiable manifolds, tangent space, vector fields, differentiable maps, 1-forms, tensors, linear connections, Riemannian manifolds, and the Levi—Civita connection.

The material of this chapter forms the basis for next chapters. Statistical manifolds are abstract generalizations of statistical models. Even if a statistical manifold is treated as a purely geometric object, however, the motivation for the definitions is inspired from statistical models. In this new framework, the manifold of density functions is replaced by an arbitrary Riemannian manifold M , and the Fisher information matrix is replaced by the Riemannian metric g of the manifold M.

The skewness tensor, which measures the cummulants of the third order on a statistical model, is replaced by a 3-covariant skewness tensor. This chapter defines the volume elements associated with two dual connections and investigates their relationship.

First, we define the Riemannian volume element and show that it is parallel with respect to the Levi—Civita connection. Since the converse is also true, this provides an alternate definition for the volume element used in defining volume elements associated with other connections.

The volume elements for the exponential model and mixture model are computed, as examples of distinguished importance in the theory. Each linear connection induces a divergence, which is used to define a Laplacian. Dual connections yield to dual Laplacians. Their relationship with Hessians, curvature vector fields, and dual volume elements is emphasized. Eguchi [ 38 , 39 , 41 ] has shown that a contrast function D induces a Riemannian metric by its second order derivatives, and a pair of dual connections by its third order derivatives.

This chapter deals with some important examples of contrastfunctions on a space of density functions, such as: Bregman divergence, Kullback—Leibler relative entropy, f -divergence, Hellinger distance, Chernoff information, Jefferey distance, Kagan divergence, and exponential contrast function. The goal of this chapter is to produce hands-on examples for the theoretical concepts introduced in Chap. This chapter studies the geometric structure induced on a submanifold by the dualistic structure of a statistical manifold.

This includes the study of the first and second fundamental forms, curvatures, mean curvatures, and the relations among them. This material adapts the well-known theory of submanifolds to the statistical manifolds framework and consists mainly in the contributions of the authors. Title Geometric Modeling in Probability and Statistics. Publisher Springer International Publishing. Print ISBN Electronic ISBN Statistics - Geometric Probability Distribution

The use of geometrical methods in statistics has a long and rich history highlighting many different aspects. These methods are usually based on a Riemannian structure defined on the space of parameters that characterize a family of probabilities. In this paper, we consider the finite dimensional case but the basic ideas can be extended similarly to the infinite-dimensional case. Our aim is to understand exponential families of probabilities on a finite set from an intrinsic geometrical point of view and not through the parameters that characterize some given family of probabilities. For that purpose, we consider a Riemannian geometry defined on the set of positive vectors in a finite-dimensional space. Geometric Modeling in Probability and Statistics

It seems that you're in Germany. We have a dedicated site for Germany. Authors: Calin , Ovidiu, Udriste , Constantin. This book covers topics of Informational Geometry, a field which deals with the differential geometric study of the manifold probability density functions. This is a field that is increasingly attracting the interest of researchers from many different areas of science, including mathematics, statistics, geometry, computer science, signal processing, physics and neuroscience. 