Prior Distributions on Measure Space

A measure is the formal representation of the non-negative additive functions that abound in science. We review and develop the art of assigning Bayesian priors to measures. Where necessary, spatial correlation is delegated to correlating kernels imposed on otherwise uncorrelated priors. The latter must be infinitely divisible (ID) and hence described by the Lévy–Khinchin representation. Thus the fundamental object is the Lévy measure, the choice of which corresponds to different ID process priors. The general case of a Lévy measure comprising a mixture of assigned base measures leads to a prior process comprising a convolution of corresponding processes. Examples involving a single base measure are the gamma process, the Dirichlet process (for the normalized case) and the Poisson process. We also discuss processes that we call the supergamma and super-Dirichlet processes, which are double base measure generalizations of the gamma and Dirichlet processes. Examples of multiple and continuum base measures are also discussed. We conclude with numerical examples of density estimation.