Statistical Structures on Metric Path Spaces

The authors extend the notion of statistical structure from Riemannian geometry to the general framework of path spaces endowed with a nonlinear connection and a generalized metric. Two particular cases of statistical data are defined. The existence and uniqueness of a nonlinear connection corresponding to these classes is proved. Two Koszul tensors are introduced in accordance with the Riemannian approach. As applications, the authors treat the Finslerian (??, ??)-metrics and the Beil metrics used in relativity and field theories while the support Riemannian metric is the Fisher-Rao metric of a statistical model.