The intrinsic geometry of statistical models

Let p(ξ) be a smooth family of probability measures depending on the parameter (Formula Presented.). Its Fisher metric is (Formula Presented.) and the Amari–Chentsov tensor is (Formula Presented) then yields two torsion free, flat connections, the exponential connection ∇(1), and the mixture connection ∇(−1), that are dual to each other w.r.t. gij. This chapter studies the intrinsic geometry resulting from such a structure. The metric is then locally generated by a strictly convex function ψ, gij= ∂i∂jψ, and its Legendre transform φ generates the inverse metric tensor. The Gibbs families of statistical mechanics are special cases of exponential families, and φ then corresponds to the free energy, ψ to the negative of the entropy. The geometry of exponential and mixture families can also be described in terms of the Kullback–Leibler divergence. More generally, we can associate a canonical divergence to a dualistic structure in the sense of Amari–Nagaoka, consisting of a metric g and a pair of torsion free connections that are dual w.r.t. g. Such a structure is equivalently given in terms of a metric tensor g and a 3-symmetric tensor T, a statistical manifold in the sense of Lauritzen. We close the circle with Lê’s embedding theorem that says that any such (not necessarily) compact statistical manifold can be embedded, preserving g and T, into the corresponding structure on some finite (discrete) sample space.

DDC Class

510: Mathematik

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The intrinsic geometry of statistical models