PRIOR DISTRIBUTIONS ON MEASURE SPACE

A measure is the formal representation of the non-negative additive fu nctions that abound in science. We review and develop the art of assig ning 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 descri bed by the Levy-Khinchin representation. Thus the fundamental object i s the Levy measure, the choice of which corresponds to different ID pr ocess priors. The general case of a Levy measure comprising a mixture of assigned base measures leads to a prior process comprising a convol ution of corresponding processes. Examples involving a single base mea sure 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 me asure generalizations of the gamma and Dirichlet processes. Examples o f multiple and continuum base measures are also discussed. We conclude with numerical examples of density estimation.