The Differential Geometric View of Statistics and Estimation

Felix Opitz
2009 Jahrestagung der Gesellschaft für Informatik  
Statistics and estimation theory is enriched with techniques derived from differential geometry. This establishes the increasing topic of information geometry. This allows new insights into these classical topics. Differential geometry offers a wide spectrum of applications within statistic inference and estimation theory. Especially, many topics of information theory can be interpreted in a geometric way, which offers new insights into this discipline. This is widely called information
more » ... . Therefore, parameterised probability densities determine manifold like structures, the so called statistic manifolds. The log-likelihood determines an embedding of this manifolds into affine spaces. The Fisher information delivers a metric for this static manifolds. Further one can define geodesics in this manifolds, which allows to measure the distance between different probability densities. Other topics are asymptotic of estimators, sufficiency of statistics, flatness, and divergence of densities and contrast functions like the Kullback-Leibler information. These topics may have also consequences in signal processing like constant false alarm rate (CFAR) and space time adaptive processing (STAP). The first section gives a short course in differential geometry. It covers both the most demonstrative extrinsic and the more abstract intrinsic view of differential geometry. The aim of the presentation is to make the analogies with information geometry visible. The second part of this paper introduces the topic of information geometry and possible application in signal processing and tracking.
dblp:conf/gi/Opitz09 fatcat:x4key2ag3bgxjm4zgefjdbgopq