On the Support of MacEachern’s Dependent Dirichlet Processes and Extensions

We study the support properties of Dirichlet process{based models for sets of predictor{dependent probability distributions. Exploiting the connection between copulas and stochastic processes, we provide an alternative deflnition of MacEachern's dependent Dirichlet processes. Based on this deflnition, we provide su-cient conditions for the full weak support of difierent versions of the process. In particular, we show that under mild conditions on the copula functions, the version where only the support points or the weights are dependent on predictors have full weak support. In addition, we also characterize the Hellinger and Kullback{Leibler support of mixtures induced by the difierent versions of the dependent Dirichlet process. A generalization of the results for the general class of dependent stick{ breaking processes is also provided.

[1]  S. MacEachern,et al.  Bayesian Nonparametric Spatial Modeling With Dirichlet Process Mixing , 2005 .

[2]  S. MacEachern,et al.  A nonparametric Bayesian model for inference in related longitudinal studies , 2005 .

[3]  D. Dunson,et al.  Nonparametric Bayes Conditional Distribution Modeling With Variable Selection , 2009, Journal of the American Statistical Association.

[4]  Peter Müller,et al.  Semiparametric Bayesian classification with longitudinal markers , 2007, Journal of the Royal Statistical Society. Series C, Applied statistics.

[5]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[6]  Ilenia Epifani,et al.  Nonparametric priors for vectors of survival functions , 2009 .

[7]  Adrian F. M. Smith,et al.  Bayesian Statistics 5. , 1998 .

[8]  Ramsés H. Mena,et al.  Geometric stick-breaking processes for continuous-time Bayesian nonparametric modeling , 2011 .

[9]  J. Sethuraman.  A CONSTRUCTIVE DEFINITION OF DIRICHLET PRIORS , 1991 .

[10]  E. T. Olsen,et al.  Copulas and Markov processes , 1992 .

[11]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[12]  Antonietta Mira,et al.  Bayesian hierarchical nonparametric inference for change-point problems , 1996 .

[13]  Andriy Norets,et al.  POSTERIOR CONSISTENCY IN CONDITIONAL DENSITY ESTIMATION BY COVARIATE DEPENDENT MIXTURES , 2011, Econometric Theory.

[14]  H. Joe.  Multivariate models and dependence concepts , 1998 .

[15]  D. Dunson,et al.  Kernel stick-breaking processes. , 2008, Biometrika.

[16]  Arnaud Doucet,et al.  Bayesian Inference for Linear Dynamic Models With Dirichlet Process Mixtures , 2007, IEEE Transactions on Signal Processing.

[17]  N. Pillai,et al.  Bayesian density regression , 2007 .

[18]  S. Ghosal,et al.  Kullback Leibler property of kernel mixture priors in Bayesian density estimation , 2007, 0710.2746.

[19]  S. Walker,et al.  Extending Doob's consistency theorem to nonparametric densities , 2004 .

[20]  S. MacEachern,et al.  An ANOVA Model for Dependent Random Measures , 2004 .

[21]  P. Müller,et al.  Bayesian curve fitting using multivariate normal mixtures , 1996 .

[22]  Andriy Norets,et al.  Approximation of conditional densities by smooth mixtures of regressions , 2010, 1010.0581.

[23]  A. N. Kolmogorov,et al.  Foundations of the theory of probability , 1960 .

[24]  Wesley O Johnson,et al.  Bayesian Nonparametric Nonproportional Hazards Survival Modeling , 2009, Biometrics.

[25]  T. Ferguson.  A Bayesian Analysis of Some Nonparametric Problems , 1973 .

[26]  M. Escobar,et al.  Bayesian Density Estimation and Inference Using Mixtures , 1995 .

[27]  T. Ferguson.  BAYESIAN DENSITY ESTIMATION BY MIXTURES OF NORMAL DISTRIBUTIONS , 1983 .

[28]  David B. Dunson,et al.  Posterior consistency in conditional distribution estimation , 2013, J. Multivar. Anal..

[29]  Maria De Iorio,et al.  Bayesian semiparametric inference for multivariate doubly-interval-censored data , 2010, 1101.1415.

[30]  Fabrizio Leisen,et al.  Vectors of two-parameter Poisson-Dirichlet processes , 2011, J. Multivar. Anal..

[31]  J. Pitman,et al.  The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator , 1997 .

[32]  Albert Y. Lo,et al.  On a Class of Bayesian Nonparametric Estimates: I. Density Estimates , 1984 .

[33]  Lancelot F. James,et al.  Gibbs Sampling Methods for Stick-Breaking Priors , 2001 .

[34]  Pietro Muliere,et al.  A bayesian predictive approach to sequential search for an optimal dose: Parametric and nonparametric models , 1993 .