Non-Parametric Priors for Functional Data and Partition Labelling Models

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-breaking

process. In this thesis some of these are examined in more detail for similarities

and differences in their stick-breaking extensions. Two broad classes of extensions

can be defined, differentiating by how the construction of functional mixture weights

are handled: one type of process views it as the product of a sequence of marginal

mixture 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 reveals

that a more fundamental difference exists. Whereas marginal labelling models nec-

essarily assign labels only at specific arguments, joint labelling models can allow for

the assignment of labels to random subsets of the domain of the function. This leads

naturally to the idea of a stochastic process based around a random partitioning of a

bounded domain, which we call the partitioned functional Dirichlet process. Here we

explicitly model the partitioning of the domain in a constrained manner, rather than

implicitly as happens in the generalized functional Dirichlet process. Comparisons

are made in terms of posterior predictive behaviour between this model, the general-

ized functional Dirichlet process and the functional Dirichlet process. We find that

the explicit modelling of the partitioning leads to more tractable computational and

more structured posterior predictive behaviour than in the generalized functional

Dirichlet process, while still offering increased flexibility over the functional Dirich-

let process. Finally, we extend the partitioned functional Dirichlet process to the

bivariate case.