Order-Based Dependent Dirichlet Processes

• Published in: Journal of the American Statistical Association 2006-03, Vol.101 (473), p.179-194
• Main Authors: Griffin, J. E, Steel, M. F. J
• Format: Article
• Language: English
• Subjects: Applications; Atoms; Autocorrelation; Bayesian analysis; Bayesian nonparametrics; Distribution theory; Exact sciences and technology; General topics; Geometry; Inference; Markov chain Monte Carlo; Mathematics; Nonparametric models; Nonparametric regression; Parametric models; Poisson process; Probability and statistics; Probability theory and stochastic processes; Random variables; Sciences and techniques of general use; Spatial modeling; Spatial models; Statistics; Steels; Stick-breaking prior; Stochastic models; Stochastic processes; Theory and Methods; Time series models; Volatility modeling
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1198/016214505000000727

In this article we propose a new framework for Bayesian nonparametric modeling with continuous covariates. In particular, we allow the nonparametric distribution to depend on covariates through ordering the random variables building the weights in the stick-breaking representation. We focus mostly on the class of random distributions that induces a Dirichlet process at each covariate value. We derive the correlation between distributions at different covariate values and use a point process to implement a practically useful type of ordering. Two main constructions with analytically known correlation structures are proposed. Practical and efficient computational methods are introduced. We apply our framework, through mixtures of these processes, to regression modeling, the modeling of stochastic volatility in time series data, and spatial geostatistical modeling.