Fisher Geometry.pdf

Nonparametric Fisher geometry is equivalent to the geometry of an infinite-dimensional sphere. This geometry is used to develop a novel Bayesian nonparametric density estimation model. The model is based on a Gaussian process (GP) prior for the square-root density function, restricted to the infinite-dimensional sphere. The Karhunen-Lo`eve (K-L) expansion of the GP prior is used to represent the square-root density function. The restriction to the sphere translates to a restriction on the eigenvalues of the GP, allowing for efficient computation using spherical Hamiltonian Monte Carlo (HMC). The resulting model is flexible and provides plausible realizations of the uncertainty in density estimation. The χ2-process density prior is introduced, which is a novel application of the χ2-process to Bayesian density estimation. The method is an alternative to Dirichlet Process Mixture Models (DPMM) and Gaussian Process Density Sampler (GPDS), with a simpler framework and more efficient computation. Key features of the model include:

Nonparametric Fisher geometry equivalent to an infinite-dimensional sphere
Gaussian process prior for the square-root density function
Karhunen-Lo`eve expansion for representation
Restriction to the infinite-dimensional sphere for efficient computation
Spherical Hamiltonian Monte Carlo (HMC) for posterior sampling
χ2-process density prior for Bayesian density estimation
* Alternative to DPMM and GPDS with simpler framework and more efficient computation.