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 $C_b^k$ with respect to appropriate reference measures. The case $k=\infty$, in which the manifolds are modelled on Fr\'{e}chet spaces, is included. The manifolds admit the Fisher-Rao metric and the dually flat geometry of Amari's $\alpha$-covariant derivatives, for all $\alpha\in R$. By construction, they are $C^\infty$-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.