Massive Inference

Sibusiso Sibisi
University of Cambridge, Cavendish Laboratory
Madingley Road, England CB3 0HE

Abstract

Bayesian estimation of a non-negative density \phi(x) over continuous x is discussed by John Skilling in these proceedings. The analysis uses a finite discretization on x, subject to the requirement that different discretizations must be consistent if they are to be representations of the same continuum limit problem. This condition effectively leads to a gamma process `prior', which becomes a Dirichlet process if unit normalization on \phi is imposed. Here we focus on problems where such normalization holds, with density estimation as our exemplar.

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

f(x) = \int dx^\prime \phi(x^\prime)K(x,x^\prime)

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.


MaxEnt 94 Abstracts / mas@mrao.cam.ac.uk