Information Geometry

Shinto Eguchi () and Osamu Komori ()
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Shinto Eguchi: Institute of Statistical Mathematic
Osamu Komori: Seikei University

Chapter Chapter 1 in Minimum Divergence Methods in Statistical Machine Learning, 2022, pp 3-17 from Springer

Abstract: Abstract In the likelihood principle, the log-likelihood functionLog likelihood function on a statistical modelStatistical model defines the maximum likelihood estimatorMaximum Likelihood Estimator (MLE). Then the Kullback-Leibler (KL) divergenceKullback-Leibler (KL) divergence is introduced with a close relationship of the maximum likelihood, which is clarified in association with an empirical Pythagorean theoremEmpirical Pythagorean theorem. As familiar statistical modelsStatistical model, we consider a normal distribution model and a contingency tableContingency tables model to explore typical aspects of information geometry. The information metricInformation metric, the e-geodesicE-geodesic, and the m-geodesicM-geodesic paths are formulated in a general framework. Finally, we observe the Pythagorean theorem in the space of all probability density functions. If the e-geodesicE-geodesic and m-geodesicM-geodesic paths orthogonally intersect with respect to the information metricInformation metric, then the two paths induce a right triangle with a Pythagorean identity in the sense of the KL divergenceKullback-Leibler (KL) divergence.

Date: 2022
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DOI: 10.1007/978-4-431-56922-0_1

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