Rho–tau embedding and gauge freedom in information geometry

The standard model of information geometry, expressed as Fisher–Rao metric and Amari-Chensov tensor, reflects an embedding of probability density by $$\log $$log-transform. The present paper studies parametrized statistical models and the induced geometry using arbitrary embedding functions, comparing single-function approaches (Eguchi’s U-embedding and Naudts’ deformed-log or phi-embedding) and a two-function embedding approach (Zhang’s conjugate rho-tau embedding). In terms of geometry, the rho-tau embedding of a parametric statistical model defines both a Riemannian metric, called “rho-tau metric”, and an alpha-family of rho-tau connections, with the former controlled by a single function and the latter by both embedding functions $$\rho $$ρ and $$\tau $$τ in general. We identify conditions under which the rho-tau metric becomes Hessian and hence the $$\pm 1$$±1 rho-tau connections are dually flat. For any choice of rho and tau there exist models belonging to the phi-deformed exponential family for which the rho-tau metric is Hessian. In other cases the rho–tau metric may be only conformally equivalent with a Hessian metric. Finally, we show a formulation of the maximum entropy framework which yields the phi-exponential family as the solution.