JE Griffin and MFJ Steel
Abstract: In this paper we propose a new framework for Bayesian nonparametric modelling 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 which 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, though mixtures of these processes, to regression modelling, the modelling of stochastic volatility in time series data and spatial geostatistical modelling.
Keywords: Bayesian nonparametrics, Markov chain Monte Carlo, Nonparametric Regression, Spatial Modelling, Stick-breaking Prior, Volatility Modelling.