February, 2007
As a generalization of the Dirichlet process 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 Dirichlet process 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.
Keywords: Dependent Dirichlet process; Blocked Gibbs sampler; Mixture model; Nonparametric Bayes; Stick-breaking process.
The manuscript is available in PDF formats.