Information geometry and asymptotic geodesics on the space of normal distributions

Abstract

The family N of n-variate normal distributions is parameterized by the cone of positive definite symmetric n × n-matrices and the n-dimensional real vector space. Equipped with the Fisher information metric, N becomes a Riemannian manifold. As such, it is diffeomorphic, but not isometric, to the Riemannian symmetric space Pos 1 (n+1, R) of unimodular positive definite symmetric (n +1)×(n +1)-matrices. As the computation of distances in the Fisher metric for n > 1 presents some difficulties, Lovrič et al. (J Multivar Anal 74:36-48, 2000) proposed to use the Killing metric on Pos 1 (n+1, R) as an alternative metric in which distances are easier to compute. In this work, we survey the geometric properties of the space N and provide a quantitative analysis of the defect of certain geodesics for the Killing metric to be geodesics for the Fisher metric. We find that for these geodesics the use of the Killing metric as an approximation for the Fisher metric is indeed justified for long distances. Keywords Gaussian distributions • Fisher metric • Cone of positive definite matrices • Symmetric spaces • Geodesics.

A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions

“…Moreover, we obtained a closed form for the Fisher–Rao distance between normals sharing the same covariance matrix using the technique of maximal invariance under the action of the affine group in Section 1.5 . We may also consider other distances different from the Fisher–Rao distance, which admits a closed-form formula: For example, the Calvo and Oller metric distance [ 19 ] (a lower bound on the Fisher–Rao distance) or the metric distance proposed in [ 82 ] (see Appendix C ) whose geodesics enjoys the asymptotic property of the Fisher–Rao geodesics [ 89 ]). The C&O distance is very well-suited for short Fisher–Rao distances while the symmetric space distance is well-tailored for large Fisher–Rao distances.…”

“…Let us emphasize that the Killing distance is not the Fisher–Rao distance but is available in closed form as an alternative metric distance between MVNs. A Fisher geodesic defect measure of a curve c is defined in [ 89 ] by where denotes the Levi–Civita connection induced by the Fisher metric. When the curve is said to be an asymptotic geodesic of the Fisher geodesic.…”

“…The multivariate Gaussian manifold can also be embedded into the SPD cone as a Riemannian symmetric space [ 82 , 89 ] by : . We have [ 82 , 95 , 96 ] (and textbook [ 97 ], Part II Chapter 10), and the symmetric space can be embedded with the Killing Riemannian metric instead of the Fisher information metric: where is a predetermined constant (e.g., 1).…”

A note on the $f$-divergences between multivariate location-scale families with either prescribed scale matrices or location parameters

“…In general, the Rao geodesic distance between multivariate Gaussian distributions is not known in closed form [10]. For the special cases of Gaussian distributions with prescribed covariance matrix or prescribed location parameter, the Rao distance is available in closed-form [23].…”

“…We may interpret the location-scale family F = {p l,P (x) : (l, P ) ∈ H d } as obtained by the action of the affine group [10] on the standard density p. The affine group Aff(R d ) = R d ⋊ GL d (R) equipped with the (outer) semidirect product:…”