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Abstract:Information geometry is a study of statistical manifolds, that is, spaces of probability distributions from a geometric perspective. Its classical information-theoretic applications relate to statistical concepts such as Fisher information, sufficient statistics, and efficient estimators. Today, information geometry has emerged as an interdisciplinary field that finds applications in diverse areas such as radar sensing, array signal processing, quantum physics, deep learning, and optimal transport. This article presents an overview of essential information geometry to initiate an information theorist, who may be unfamiliar with this exciting area of research. We explain the concepts of divergences on statistical manifolds, generalized notions of distances, orthogonality, and geodesics, thereby paving the way for concrete applications and novel theoretical investigations. We also highlight some recent information-geometric developments, which are of interest to the broader information theory community.

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Abstract:Information geometry is the study of statistical models from a Riemannian geometric point of view. The Fisher information matrix plays the role of a Riemannian metric in this framework. This tool helps us obtain Cram\'{e}r-Rao lower bound (CRLB). This chapter summarizes the recent results which extend this framework to more general Cram\'{e}r-Rao inequalities. We apply Eguchi's theory to a generalized form of Czsisz\'ar $f$-divergence to obtain a Riemannian metric that, at once, is used to obtain deterministic CRLB, Bayesian CRLB, and their generalizations.

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Abstract:The relative $\alpha$-entropy is the R\'enyi analog of relative entropy and arises prominently in information-theoretic problems. Recent information geometric investigations on this quantity have enabled the generalization of the Cram\'{e}r-Rao inequality, which provides a lower bound for the variance of an estimator of an escort of the underlying parametric probability distribution. However, this framework remains unexamined in the Bayesian framework. In this paper, we propose a general Riemannian metric based on relative $\alpha$-entropy to obtain a generalized Bayesian Cram\'{e}r-Rao inequality. This establishes a lower bound for the variance of an unbiased estimator for the $\alpha$-escort distribution starting from an unbiased estimator for the underlying distribution. We show that in the limiting case when the entropy order approaches unity, this framework reduces to the conventional Bayesian Cram\'{e}r-Rao inequality. Further, in the absence of priors, the same framework yields the deterministic Cram\'{e}r-Rao inequality.

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Abstract:We study the geometry of probability distributions with respect to a generalized family of Csisz\'ar $f$-divergences. A member of this family is the relative $\alpha$-entropy which is also a R\'enyi analog of relative entropy in information theory and known as logarithmic or projective power divergence in statistics. We apply Eguchi's theory to derive the Fisher information metric and the dual affine connections arising from these generalized divergence functions. The notion enables us to arrive at a more widely applicable version of the Cram\'{e}r-Rao inequality, which provides a lower bound for the variance of an estimator for an escort of the underlying parametric probability distribution. We then extend the Amari-Nagaoka's dually flat structure of the exponential and mixer models to other distributions with respect to the aforementioned generalized metric. We show that these formulations lead us to find unbiased and efficient estimators for the escort model.

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