Probability density estimation on a high-dimensional space using random tessellations

Probability density estimation on a high dimensional observation space can be performed in a variety of ways. The framework that is developed is to construct a tessellation of the d-dimensional support for a probability density using some or all of the available d-dimensional observations as vertices of the tessellation. The elements of the tessellation are d-dimensional Delaunay minimum tiles.;A class of probability density estimators is then developed on the d-dimensional tessellation. This class of estimators is shown to be consistent, provided the support for the probability density is convex and all elements of the tessellation have a positive probability estimate, given a set of observations.;An algorithm is then presented that constructs a d-dimensional tessellation of Delaunay minimum tiles in an efficient manner. An algorithm is then presented for computing a probability density estimate given a d-dimensional tessellation.;The framework for probability density estimation that is developed is less data intensive than binning methods. It is also more computationally tractable than the kernel density estimation framework on high dimensional observation spaces.