The hyper-Dirichlet process and its discrete approximations: The butterfly model

• Published in: Journal of multivariate analysis 2006-04, Vol.97 (4), p.895-924
• Main Authors: Asci, C., Nappo, G., Piccioni, M.
• Format: Article
• Language: English
• Subjects: Bayesian statistics; Convergence; Coupling; Dirichlet laws and processes; Dirichlet laws and processes Bayesian statistics Markov distributions Weak convergence Coupling; Exact sciences and technology; Limit theorems; Markov distributions; Mathematical models; Mathematics; Multivariate analysis; Nonparametric inference; Parametric inference; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Statistics; Stochastic processes; Studies; Weak convergence
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1016/j.jmva.2005.08.009

The aim of this paper is the study of some random probability distributions, called hyper-Dirichlet processes. In the simplest situation considered in the paper these distributions charge the product of three sample spaces, with the property that the first and the last component are independent conditional to the middle one. The law of the marginals on the first two and on the last two components are specified to be Dirichlet processes with the same marginal parameter measure on the common second component. The joint law is then obtained as the hyper-Markov combination, introduced in [A.P. Dawid, S.L. Lauritzen, Hyper-Markov laws in the statistical analysis of decomposable graphical models, Ann. Statist. 21 (3) (1993) 1272–1317], of these two Dirichlet processes. The processes constructed in this way in fact are in fact generalizations of the hyper-Dirichlet laws on contingency tables considered in the above paper. Our main result is the convergence to the hyper-Dirichlet process of the sequence of hyper-Dirichlet laws associated to finer and finer “discretizations” of the two parameter measures, which is proved by means of a suitable coupling construction.