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an elementary introduction to information geometry

Decades later, Amari and his collaborators [2,7] developed the dual geometric structures, which are considered as the core of parametrised information geometry; for more, see [9,26,36. extended class of Chi-type distances. By considering the geometry of the underlying statistical manifold, we define exactly the solution of the optimization problem as the unique intersection of a geodesic with a dual hyperplane. A set of observations whose expectations are linear functions of a number of unknown parameters being given, the problem which Markoff posed for solution is to find out a linear function of observations whose expectation is an assigned linear function of the unknown parameters and whose variance is a minimum. We give the geometric structure of the power inverse Gaussian manifold from the viewpoint of information geometry. Standard properties of ϕ-divergences of probability measures are widely applied in various areas of information processing. Appendices. 623-656 (October). Information-theoreticmeasures, such as the entropy, the cross-entropy and the Kullback-Leibler divergence between two mixture models, are core primitives in many signal processing tasks. ONE-PARAMETER CURVED EXPONENTIAL FAMILIES. Considering the Fisher information metric as a Riemannian metric, information geometry was developed to understand the intrinsic properties of statistical models, which play important roles in statistical inference, etc. In pattern recognition, patch matching is a fundamental task that is time consuming, specially when zoom factors and symmetries are handled. The differential-geometric structure of the set of positive densities on a We provide new results and extend previous resuts on the robustness Une partie du travail antérieur est ensuite esquissée. ... A modern and broad definition of information geometry can be stated. Among the desirable supplementary properties facilitating employment of mathematical methods is the metricity of ϕ-divergences, or the metricity of their powers. The space of Gaussian measures is of finite dimension, In the second part of the paper, we consider the Bhattacharyya distance that is commonly used to measure overlapping degree of probability distributions. Finally, we show how to compute statistical H\"older centroids with respect to those divergences, and carry out center-based clustering toy experiments on a set of Gaussian distributions that demonstrate empirically that symmetrized H\"older divergences outperform the symmetric Cauchy-Schwarz divergence. We demonstrate our approach with a clustering task on a mixture family manifold. $$ The positivity of (g ij ) makes S to be a Riemannian manifold; see e.g. Physica A: Statistical Mechanics and its Applications, An Elementary Introduction to Information Geometry, Statistical Einstein manifolds of exponential families with group-invariant potential functions, Fisher-Rao Geometry and Jeffreys Prior for Pareto Distribution, Optimal Pilot Design for MIMO Broadcasting Systems Based on the Positive Definite Matrix Manifold, A Matrix Information-Geometric Method for Change-Point Detection of Rigid Body Motion, Target Detection Algorithm with Information Geometry under Cooperative Position, Information geometry and phase transitions, Information Geometry: An Introduction to New Models for Signal Processing. Second-Order Asymptotics. [1,2,10, ... To deal with the inference problem after we observed the data x = (x 1 , . its completion as a metric space provides a complete picture In these cases, the dually flat construction remains theoretical and cannot be used by information-geometric algorithms. To circumvent this difficulty, the Chernoff Upper Bound (CUB) for larger SNR and the Fisher information at low SNR are derived and studied, based on information geometry theory. d \mapsto f'(x, d): = \mathop {\lim }\limits_{t \downarrow 0} \frac{{f(x + td) - f(x)}} There is no assumption about the distribution of the observations except that each has a finite variance. A divergence function on a manifold M defines a Riemannian metric g and dually coupled affine connections ∇ and ∇* on M. When M is dually flat, that is flat with respect to ∇ and ∇*, a canonical divergence is known, which is uniquely determined from (M, g, ∇, ∇*). 2020; 22(10):1100. The performance of our method was evaluated by numerical examples and manipulator experiments. We also show that Kullback-Leibler Furthermore, we prove that the skew Jensen-Shannon statistical divergence between w-mixtures amount to skew Jensen divergences on their parameters and state several divergence inequalities between w-mixtures and their closures. We extract a local component from a large neural system, and define its relative Fisher information metric that describes accurately this small component, and is invariant to the other parts of the system. In particular, the integral is bounded by O(n 4.5 m) for combi-natorial linear programs including network flow problems where m is the number of constraints. In this chapter, we review how such divergence functions induce (i) a statistical structure (i.e., a Riemannian metric with a pair of conjugate affine connections) on \(\mathfrak {M}\); (ii) a symplectic structure on \(\mathfrak {M}\times \mathfrak {M}\) if they are “proper”; (iii) a Kähler structure on \(\mathfrak {M}\times \mathfrak {M}\) if they further satisfy a certain condition. We use cookies on our website to ensure you get the best experience. Exponential Family Regression and Diagnostics. We prove that its geometrical structure is isometric to the Poincar\'e upper half-plane model, and then study the corresponding geometrical features by presenting explicit expressions for connection, curvature and geodesics. Assuming that the users have the channels’ statistical information and adopt the minimum mean square error (MMSE) receivers, this article studies how to design an optimal training sequence to minimize the total weighted mean square error (MSE) of channel estimation. The tightest CUB is reached for the value minimizing the error exponent, denoted by s ⋆ . The exposition is self-contained by concisely introducing the necessary concepts of differential geometry. Please let us know what you think of our products and services. We define several types of Bregman diagrams, establish correspondences between those diagrams (using the Legendre transformation), and show how to compute them efficiently. Moreover, Thus we get an efficient algorithm for computing the Bhattacharyya centroid of a set of parametric distributions belonging to the same exponential families, improving over former specialized methods found in the literature that were limited to univariate or “diagonal” multivariate Gaussians.

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