The Dirichlet process has a discrete representation: to any desired accuracy, a sample from the process consists of a finite number of isolated point masses even in the continuum limit. This evades the Law of Large Numbers which would otherwise lead to flat, zero-variance estimates. Because of this property, we term the use of such a process Massive Inference.
In density estimation, we are given a set of iidobservations drawn from an unknown continuous density f from which we wish to estimate f. In this problem, as in many others, f is presumed to be smooth. We incorporate smoothness through the integral formulation
By construction, \phi is a latent density containing no spatial smoothness---the appropriate smoothness being delegated to the kernel function K, which is assigned at least to within a few shape parameters. We use a Dirichlet process prior on \phi so that the above integral effectively becomes a finite discrete sum over the point masses of \phi.
We present numerical investigations of textbook data in one and two dimensions.