Manifolds of Differentiable Densities

We develop a family of infinite-dimensional (i.e. non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class Ck b with respect to appropriate reference measures. The case k = ∞, in which the manifolds are modelled on Fréchet spaces, is included. The manifolds admit the Fisher-Rao metric and the dually flat geometry of Amari’s α-covariant derivatives, for all α ∈ R. By construction, they are C-embedded submanifolds of particular manifolds of finite measures. Unusually for the non-parametric case, the likelihood function associated with a finite sample is a continuous function on each of the manifolds.