Biometrika Advance Access originally published online on April 30, 2008
Biometrika 2008 95(2):307-323; doi:10.1093/biomet/asn012
© US Government/Department of Health and Human Services 2008; Published by the Biometrika Trust
Articles |
Kernel stick-breaking processes
Biostatistics Branch, National Institute of Environmental Health Sciences, P.O. Box 12233, Research Triangle Park, North Carolina 27709, U.S.A. dunson1{at}niehs.nih.gov
Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, U.S.A. parkj3{at}niehs.nih.gov
Received for publication 1 November 2006. Revision received 1 August 2007.
We propose a class of kernel stick-breaking processes for uncountable collections of dependent random probability measures. The process is constructed by first introducing an infinite sequence of random locations. Independent random probability measures and beta-distributed random weights are assigned to each location. Predictor-dependent random probability measures are then constructed by mixing over the locations, with stick-breaking probabilities expressed as a kernel multiplied by the beta weights. Some theoretical properties of the process are described, including a covariate-dependent prediction rule. A retrospective Markov chain Monte Carlo algorithm is developed for posterior computation, and the methods are illustrated using a simulated example and an epidemiological application.
Key Words: Conditional density estimation • Dependent Dirichlet process • Kernel methods • Nonparametric Bayes • Mixture model • Prediction rule • Random partition
References
-
Aldous D. J. Exchangeability and related topics. In: École d'Été de Probabilités de Saint-Flour XII—Hennequin P. L., ed. (1985) Berlin: Springer. 1–198. Lecture Notes in Mathematics 1117.
Blackwell D., Macqueen J. B. Ferguson distributions via Pólya urn schemes. Ann. Statist. (1973) 1:353–5.[CrossRef]
Barry D., Hartigan J. A. Product partition models for change point problems. Ann. Statist. (1992) 20:260–79.[CrossRef]
Caron F., Davy M., Doucet A., Duflos E., Vanheeghe P. Bayesian inference for dynamic models with Dirichlet process mixtures. In: International Conference on Information Fusion (2006) Florence, Italy: INRIA–CCSd–CNRS. 1–8.
Cifarelli D. M., Regazinni E. Nonparametric statistical problems under partial exchangeability: the use of associative means. Ann. Inst. Mat. Finian. Univ. Torino, II (1978) 12:1–36.
De Iorio M., Müller P., Rosner G. L., Maceachern S. N. An ANOVA model for dependent random measures. J. Am. Statist. Assoc. (2004) 99:205–15.[CrossRef][Web of Science]
Dunson D. B. Bayesian dynamic modelling of latent trait distributions. Biostatistics (2006) 7:551–68.
Dunson D. B., Herring A. H., Engel S. M. Bayesian selection and clustering of polymorphisms in functionally-related genes. J. Am. Statist. Assoc. (2007) forthcoming.
Dunson D. B., Pillai N., Park J-H. Bayesian density regression. J. R. Statist. Soc. B (2007) 69:163–83.[CrossRef]
Ferguson T. S. A Bayesian analysis of some nonparametric problems. Ann. Statist. (1973) 1:209–30.[CrossRef]
Ferguson T. S. Prior distributions on spaces of probability measures. Ann. Statist. (1974) 2:615–29.[CrossRef]
Gelfand A. E., Kottas A., Maceachern S. N. Bayesian nonparametric spatial modelling with Dirichlet process mixing. J. Am. Statist. Assoc. (2005) 100:1021–35.[CrossRef][Web of Science]
Griffin J. E., Steel M. F. J. Order-based dependent Dirichlet processes. J. Am. Statist. Assoc. (2006) 101:179–94.[CrossRef][Web of Science]
Ishwaran H., James L. F. Gibbs sampling methods for stick-breaking priors. J. Am. Statist. Assoc. (2001) 96:161–73.[CrossRef][Web of Science]
Ishwaran H., James L. F. Generalized weighted Chinese restaurant processes for species sampling mixture models. Statist. Sinica (2003) 13:1211–35.
Ishwaran H., Zarepour M. Markov chain Monte Carlo in approximate Dirichlet and beta two-parameter process hierarchical models. Biometrika (2000) 87:371–90.
Kim S., Tadesse M. G., Vannucci M. Variable selection in clustering via Dirichlet process mixture models. Biometrika (2006) 93:877–93.
Longnecker M. P., Klebanoff M. A., Zhou H. B., Brock J. W. Association between maternal serum concentration of the ddt metabolite dde and preterm and small-for-gestational-age babies at birth. Lancet (2001) 358:110–4.[CrossRef][Web of Science][Medline]
Maceachern S. N. Estimating normal means with a conjugate style Dirichlet process prior. Commun. Statist. B (1994) 23:727–41.
Maceachern S. N. Dependent onparametric processes. In: Proc. Bayesian Statist. Sci. Sect. (1999) Alexandria, VA: American Statistical Association. 50–5.
Maceachern S. N. Decision theoretic aspects of dependent nonparametric processes. In: Bayesian Methods With Applications to Science, Policy, and Official Statistics—George E., ed. (2001) Crete: International Society for Bayesian Analysis. 551–60.
Medvedovic M., Yeung K. Y., Bumgarner R. E. Bayesian mixture model based clustering of replicated microarray data. Bioinformatics (2004) 20:1222–32.
Müller P., Quintana F., Rosner G. A method for combining inference across related nonparametric Bayesian models. J. R. Statist. Soc. B (2004) 66:735–49.[CrossRef]
Papaspiliopoulos O., Roberts G. O. Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models. Biometrika (2008) 95:169–86.
Pennell M. L., Dunson D. B. Bayesian semiparametric dynamic frailty models for multiple event time data. Biometrics (2006) 62:1044–52.[CrossRef][Web of Science][Medline]
Pitman J. Some developments of the Blackwell-MacQueen urn scheme. In: Statistics, Probability and Game Theory—Ferguson T. S., Shapley L. S., MacQueen J. B., eds. (1996) 30. Hayward, CA: Inst. Math. Statist. 245–67. IMS Lecture Notes–Monograph Series.
Quintana F. A., Iglesias P. L. Bayesian clustering and product partition models. J. R. Statist. Soc. B (2003) 65:557–74.[CrossRef]
Sethuraman J. A constructive definition of Dirichlet priors. Statist. Sinica (1994) 4:639–50.
West M., Müller P., Escobar M. D. Hierarchical priors and mixture models, with applications in regression and density estimation. In: Aspects of Uncertainty: A Tribute to D.V. Lindley—Smith A. F. M., Freeman P. R., eds. (1994) New York: John Wiley. 363–86.
CiteULike 
Connotea 
Del.icio.us What's this?
| ||||||||||||||||||||||||||||||||||||||||||||||||||