Nonparametric Fisher Geometry with Application to Density Estimation
Andrew Holbrook, Shiwei Lan, Jeffrey Streets, Babak Shahbaba
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 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. As discussed here, this insight leads to a novel Bayesian nonparametric density estimation model.
BibTeX
@InProceedings{pmlr-v124-holbrook20a,
title = {Nonparametric Fisher Geometry with Application to Density Estimation},
author = {Holbrook, Andrew and Lan, Shiwei and Streets, Jeffrey and Shahbaba, Babak},
booktitle = {Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI)},
pages = {101--110},
year = {2020},
editor = {Peters, Jonas and Sontag, David},
volume = {124},
series = {Proceedings of Machine Learning Research},
month = {03--06 Aug},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v124/holbrook20a/holbrook20a.pdf},
url = {https://proceedings.mlr.press/v124/holbrook20a.html},
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 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. As discussed here, this insight leads to a novel Bayesian nonparametric density estimation model.}
}