Non-Parametric Priors for Functional Data and Partition Labelling Models

Abstract

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