The nonparametric Fisher geometry and the chi-square process density prior

Abstract: 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.