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The local Dirichlet process

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Abstract

As a generalization of the Dirichlet process (DP) to allow predictor dependence, we propose a local Dirichlet process (lDP). The lDP provides a prior distribution for a collection of random probability measures indexed by predictors. This is accomplished by assigning stick-breaking weights and atoms to random locations in a predictor space. The probability measure at a given predictor value is then formulated using the weights and atoms located in a neighborhood about that predictor value. This construction results in a marginal DP prior for the random measure at any specific predictor value. Dependence is induced through local sharing of random components. Theoretical properties are considered and a blocked Gibbs sampler is proposed for posterior computation in lDP mixture models. The methods are illustrated using simulated examples and an epidemiologic application.

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References

  • Beal, M., Ghahramani, Z., Rasmussen, C. (2002). The infinite hidden Markov model. In Neural information processing systems (Vol. 14). Cambridge: MIT Press.

  • Blei, D., Griffiths, T., Jordan, M., Tenenbaum, J. (2004). Hierarchical topic models and the nested Chinese restaurant process. In Neural information processing systems (Vol. 16). Cambridge: MIT Press.

  • Caron, F., Davy, M., Doucet, A., Duflos, E., Vanheeghe, P. (2006). Bayesian inference for dynamic models with Dirichet process mixtures. In International conference on information fusion, Italia, July 10–13.

  • De Iorio M., Müller P., Rosner G.L., MacEachern S.N. (2004) An Anova model for dependent random measures. Journal of the American Statistical Association 99: 205–215

    Article  MATH  MathSciNet  Google Scholar 

  • Dowse K.G., Zimmet P.Z., Alberti G.M.M., Bringham L., Carlin J.B., Tuomlehto J., Knight L.T., Gareeboo H. (1993) Serum insulin distributions and reproducibility of the relationship between 2-hour insulin and plasma glucose levels in Asian Indian, Creole, and Chinese Mauritians. Metabolism 42: 1232–1241

    Article  Google Scholar 

  • Duan, J. A., Guidani, M., Gelfand, A. E. (2005). Generalized spatial Dirichlet process models. ISDS Discussion Paper, 05-23, Durham: Duke University.

  • Dunson D.B. (2006) Bayesian dynamic modeling of latent trait distributions. Biostatistics 7: 551–568

    Article  MATH  Google Scholar 

  • Dunson D.B., Park J.-H. (2008) Kernel stick-breaking process. Biometrika 95: 307–323

    Article  MATH  MathSciNet  Google Scholar 

  • Dunson D.B., Peddada S.D. (2008) Bayesian nonparametric inference on stochastic ordering. Biometrika 95: 859–874

    Article  MATH  MathSciNet  Google Scholar 

  • Dunson D.B., Pillai N., Park J.-H. (2007) Bayesian density regression. Journal of the Royal Statistical Society, Series B 69: 163–183

    Article  MATH  MathSciNet  Google Scholar 

  • Escobar M.D. (1994) Estimating normal means with a Dirichlet process prior. Journal of the American Statistical Association 89: 268–277

    Article  MATH  MathSciNet  Google Scholar 

  • Escobar M.D., West M. (1995) Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association 90: 577–588

    Article  MATH  MathSciNet  Google Scholar 

  • Ferguson T.S. (1973) A Bayesian analysis of some nonparametric problems. The Annals of Statistics 1: 209–230

    Article  MATH  MathSciNet  Google Scholar 

  • Ferguson T.S. (1974) Prior distributions on spaces of probability measures. The Annals of Statistics 2: 615–629

    Article  MATH  MathSciNet  Google Scholar 

  • Fraley C., Raftery A.E. (2002) Model-based clustering, discriminant analysis, and density estimation. Journal of the American Statistical Association 97: 611–631

    Article  MATH  MathSciNet  Google Scholar 

  • Gelfand A.E., Kottas A., MacEachern S.N. (2004) Bayesian nonparametric spatial modeling with Dirichlet process mixing. Journal of the American Statistical Association 100: 1021–1035

    Article  MathSciNet  Google Scholar 

  • Ghosal S., Vander Vaart A.W. (2007) Posterior convergence rates of Dirichlet mixtures at smooth densities. The Annals of Statistics 35(2): 697–723

    Article  MATH  MathSciNet  Google Scholar 

  • Ghosal S., Ghosh J.K., Ramamoorthi R.V. (1999) Posterior consistency of Dirichlet mixtures in density estimation. The Annals of Statistics 27: 143–158

    Article  MATH  MathSciNet  Google Scholar 

  • Griffin J.E., Steel M.F.J. (2006) Order-based dependent Dirichlet processes. Journal of the American Statistical Association 101: 179–194

    Article  MATH  MathSciNet  Google Scholar 

  • Griffin, J. E., Steel, M. F. J. (2008). Bayesian nonparametric modeling with the Dirichlet process regression smoother. Technical Report, University of Warwick.

  • Ishwaran H., James L.F. (2001) Gibbs sampling methods for stick-breaking priors. Journal of the American Statistical Association 96: 161–173

    Article  MATH  MathSciNet  Google Scholar 

  • Kim S., Tadesse M.G., Vannucci M. (2006) Variable selection in clustering via Dirichlet process mixture models. Biometrika 94: 877–893

    Article  MathSciNet  Google Scholar 

  • Lijoi A., Prünster I., Walker S.G. (2005) On consistency of non-parametric normal mixtures for Bayesian density estimation. Journal of the American Statistical Association 100: 1292–1296

    Article  MATH  MathSciNet  Google Scholar 

  • Lo A.Y. (1984) On a class of Bayesian nonparametric estimates: I. Density estimates. The Annals of Statistics 12: 351–357

    MATH  Google Scholar 

  • MacEachern, S. N. (1999). Dependent nonparametric processes. In ASA proceedings of the section on bayesian statistical science. Alexandria: American Statistical Association.

  • MacEachern, S. N. (2000). Dependent Dirichlet processes. Unpublished manuscript, Department of Statistics, The Ohio State University.

  • MacEachern S.N. (2001) Decision theoretic aspects of dependent nonparametric processes. In: George E. (eds) Bayesian methods with applications to science, policy and official statistics. ISBA, Creta, pp 551–560

    Google Scholar 

  • Müller P., Quintana F., Rosner G. (2004) A method for combining inference across related nonparametricBayesian models. Journal of the Royal Statistical Society B 66: 735–749

    Article  MATH  Google Scholar 

  • Pennell M.L., Dunson D.B. (2006) Bayesian semiparametric dynamic frailty models for multiple event time data. Biometrics 62: 1044–1052

    Article  MATH  MathSciNet  Google Scholar 

  • Pitman J. (1995) Exchangeable and partially exchangeable random partitions. Probability Theory and Related Fields 102: 145–158

    Article  MATH  MathSciNet  Google Scholar 

  • Pitman, J. (1996). Some developments of the Blackwell-MacQueen urn scheme. In T. S. Ferguson, L. S. Shapley, J. B. MacQueen (Eds.), Statistics, probability and game theory. IMS Lecture Notes-Monograph Series (Vol. 30, pp. 245–267), Hayward: Institute of Mathematical Statistics.

  • Quintana F.A. (2006) A predictive view of Bayesian clustering. Journal of Statistical Planning and Inference 136: 2407–2429

    Article  MATH  MathSciNet  Google Scholar 

  • Quintana F.A., Iglesias P.L. (2003) Bayesian Clustering and product partition models. Journal of the Royal Statistical Society B 65: 557–574

    Article  MATH  MathSciNet  Google Scholar 

  • Sethuraman J. (1994) A constructive definition of the Dirichlet process prior. Statistica Sinica 2: 639–650

    MathSciNet  Google Scholar 

  • Smith, J. W., Everhart, J. E., Dickson, W. C., Knowler, W. C., Johannes, R. S. (1988). Using the ADAP learning algorithm to forecast the onset of diabetes mellitus. In Proceedings of the symposium on computer applications in medical care, pp. 261–265.

  • Xing, E. P., Sharan, R., Jordan, M. (2004). Bayesian haplotype inference via the Dirichlet process. In Proceedings of the international conference on machine learning (ICML).

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Authors and Affiliations

  1. Department of Biostatistics, Harvard School of Public Health, 655 Huntington Ave. Bldg 2, Room 435A, Boston, MA, 02115, USA

    Yeonseung Chung

  2. Department of Statistical Science, Duke University, 219A Old Chemistry Bldg, Box 90251, Durham, NC, 27708, USA

    David B. Dunson

Authors
  1. Yeonseung Chung
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  2. David B. Dunson
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Correspondence to Yeonseung Chung.

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Chung, Y., Dunson, D.B. The local Dirichlet process. Ann Inst Stat Math 63, 59–80 (2011). https://doi.org/10.1007/s10463-008-0218-9

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  • DOI: https://doi.org/10.1007/s10463-008-0218-9

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