  It is well known that the Fisher information induces a Riemannian geometry on
parametric families of probability density functions. Following recent work, we
consider the nonparametric generalization of the Fisher geometry. The resulting
nonparametric Fisher geometry is shown to be equivalent to a familiar, albeit
infinite-dimensional, geometric object---the sphere. By shifting focus away
from density functions and toward \emph{square-root} density functions, one may
calculate theoretical quantities of interest with ease. More importantly, the
sphere of square-root densities is much more computationally tractable. This
insight leads to a novel Bayesian nonparametric density estimation model. We
construct the $\chi^2$-process density prior by modeling the square-root
density with a restricted Gaussian process prior. Inference over square-root
densities is fast, and the model retains the flexibility characteristic of
Bayesian nonparametric models. Finally, we formalize the relationship between
spherical HMC in the infinite-dimensional limit and standard Riemannian HMC.
