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Choosing the Fisher information as the metric tensor for a Riemannianmanifold provides a powerful yet fundamental way to understand statisticaldistribution families. Distances along this manifold become a compellingmeasure of statistical distance, and paths of shorter distance improve samplingtechniques that leverage a sequence of distributions in their operation.Unfortunately, even for a distribution as generally tractable as themultivariate normal distribution, this information geometry proves unwieldyenough that closed-form solutions for shortest-distance paths or their lengthsremain unavailable outside of limited special cases. In this review we presentfor general statisticians the most practical aspects of the Fisher geometry forthis fundamental distribution family. Rather than a differential geometrictreatment, we use an intuitive understanding of the covariance-inducedcurvature of this manifold to unify the special cases with known closed-formsolution and review approximate solutions for the general case. We also use themultivariate normal information geometry to better understand the paths ordistances commonly used in statistics (annealing, Wasserstein). Given theunavailability of a general solution, we also discuss the methods used fornumerically obtaining geodesics in the space of multivariate normals,identifying remaining challenges and suggesting methodological improvements.

The Fisher Geometry and Geodesics of the Multivariate Normals, without Differential Geometry