Non-Parametric Priors for Functional Data and Partition Labelling Models

Abstract

Previous papers introduced a variety of extensions of the Dirichlet process to the func-tional domain, focusing on the challenges presented by extending the stick-breakingprocess. In this thesis some of these are examined in more detail for similaritiesand differences in their stick-breaking extensions. Two broad classes of extensionscan be defined, differentiating by how the construction of functional mixture weightsare handled: one type of process views it as the product of a sequence of marginalmixture weights, whereas the other specifies a joint mixture weight for an entire ob-servation. These are termed “marginal” and “joint” labelling processes respectively,and we show that there are significant differences in their posterior predictive perfor-mance. Further investigation of the generalized functional Dirichlet process revealsthat a more fundamental difference exists. Whereas marginal labelling models nec-essarily assign labels only at specific arguments, joint labelling models can allow forthe assignment of labels to random subsets of the domain of the function. This leadsnaturally to the idea of a stochastic process based around a random partitioning of abounded domain, which we call the partitioned functional Dirichlet process. Here weexplicitly model the partitioning of the domain in a constrained manner, rather thanimplicitly as happens in the generalized functional Dirichlet process. Comparisonsare made in terms of posterior predictive behaviour between this model, the general-ized functional Dirichlet process and the functional Dirichlet process. We find thatthe explicit modelling of the partitioning leads to more tractable computational and more structured posterior predictive behaviour than in the generalized functionalDirichlet process, while still offering increased flexibility over the functional Dirich-let process. Finally, we extend the partitioned functional Dirichlet process to thebivariate case.