The Differential Geometric View of Statistics and Estimation

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 geometry. 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. 1 Differential geometry This section gives an short overview about differential geometry. Differential geometry may be formulated in the intrinsic and extrinsic way. Extrinsic means, that one considers a manifold (e.g. a surface) which is embedded into a larger space (e.g. the three dimensional affine space R). It was Gauss who found out that some geometric invariants of such a surface (e.g. the Gaussian curvature) depends only on the surface itself. This is the content of his famous theorema egregium. Therefore, todays intrinsic view on differential geometry is the prefered one. However, the classical extrinsic formulation has often the advantage to be more demonstrative. 1.1 Manifolds, embeddings and coordinate charts To motivate the term manifold one may think about a (orientable) surface S embedded into the three dimensional space R. The extrinsic view [Fra97] considers local parameterisations of this surface S within its circumjacent space, i.e. a smooth mapping s : U −→ R (1) s : (u, u) 7−→ s(u, u) (2) where U ∈ R is an open set and ∂s ∂u1 × ∂s ∂u2 6= 0. The intrinsic view depends on the fact that each point of the surface possesses a neighborhood which look like a open set in the two dimensional space R. This means that there is a smooth bijective mapping of this neighborhood into the R. In the general mathematical theory one defines [Jos02], [Nak90], [SGL93]: Definition 1.1 A manifold M of dimension m is a connected space1 for which every point p ∈ M has a neighborhood V ⊂ M that is homeomorph to an open subset U of R. A homeomorphism φ =  u1 ... u  : V → U (3) is called a coordinate chart. In differential geometry the manifolds carry an additional differential structure. Therefore two coordinate charts φ and ψ are called compatible, if their combinations ψ ◦ φ−1 and φ ◦ ψ−1 are smooth. A maximal set of compatible charts is called an atlas. This definition does not depend on any embeddings. Nevertheless, Whitney showed that any manifold can be embedded into an affine space of sufficient dimension.