Parametric Models for A sub n: Splitting Processes and Mixtures
Abstract
A class of parametric models, called splitting processes, is defined, using de Finetti's concept of adherent mass. Such splitting processes give rise to complex mixtures of distributions. It is proved that the nonparametric Bayesian predictive procedure, A sub n, of Hill (1968), holds exactly for a member of this class called a nested splitting process. It is also shown that the generalization of A sub n, called H sub n, to deal with ties, can hold exactly. A multivariate version of A sub n, based upon the splitting processes, is proposed. Some general considerations concerning ties and adherent masses are discussed, as well as their connection with the Dirichlet process. These include the phenomenon by which in the Dirichlet process, the posterior predictive mass builds up at the observed points, while under A sub n no mass is given to the observed points, and under H sub n some but not necessarily all posterior predictive mass builds up at the observed points. A very general class of splitting processes is then defined, which allows for some of the adherent mass at a point to be replaced by an exact tie. It is proved that both the Dirichlet process of Ferguson and A sub n can arise as different special cases of this general model. (kr)
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 24, 1989
- Accession Number
- ADA218421
Entities
People
- Bruce M. Hill
Organizations
- University of Michigan