Posterior Analysis for Normalized Random Measures with Independent Increments

One of the main research areas in Bayesian Nonparametrics is the proposal and study of priors which generalize the Dirichlet process. In this paper, we provide a comprehensive Bayesian non-parametric analysis of random probabilities which are obtained by normalizing random measures with independent...

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Bibliographic Details
Published in:Scandinavian journal of statistics 2009-03, Vol.36 (1), p.76-97
Main Authors: JAMES, LANCELOT F., LIJOI, ANTONIO, PRÜNSTER, IGOR
Format: Article
Language:English
Subjects:
Bayesian Nonparametrics
Density
Density estimation
Dirichlet process
Distribution theory
Exact sciences and technology
General topics
Inference
Integers
Mathematics
Multilevel models
Nonparametric models
normalized random measure
Poisson random measure
posterior distribution
predictive distribution
Probabilities
Probability and statistics
Probability theory and stochastic processes
Random variables
Sampling distributions
Sciences and techniques of general use
Statistical theories
Statistics
Stochastic processes
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Summary:One of the main research areas in Bayesian Nonparametrics is the proposal and study of priors which generalize the Dirichlet process. In this paper, we provide a comprehensive Bayesian non-parametric analysis of random probabilities which are obtained by normalizing random measures with independent increments (NRMI). Special cases of these priors have already shown to be useful for statistical applications such as mixture models and species sampling problems. However, in order to fully exploit these priors, the derivation of the posterior distribution of NRMIs is crucial: here we achieve this goal and, indeed, provide explicit and tractable expressions suitable for practical implementation. The posterior distribution of an NRMI turns out to be a mixture with respect to the distribution of a specific latent variable. The analysis is completed by the derivation of the corresponding predictive distributions and by a thorough investigation of the marginal structure. These results allow to derive a generalized Blackwell-MacQueen sampling scheme, which is then adapted to cover also mixture models driven by general NRMIs.
ISSN:0303-6898
1467-9469
DOI:10.1111/j.1467-9469.2008.00609.x