Non-Parametric Priors for Functional Data and Partition Labelling Models

Non-Parametric Priors for Functional Data and Partition Labelling Models by Christoph Hellmayr Department of Statistical Science Duke University Date: Approved: Alan Gelfand, Supervisor Surya Tokdar Li Ma Kingshuk Roy Choudhury An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Statistical Science in the Graduate School of Duke University 2017 Copyright c © 2017 by Christoph Hellmayr All rights reserved except the rights granted by the Creative Commons Attribution-Noncommercial Licence Abstract Previous papers introduced a variety of extensions of the Dirichlet process to the functional 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 observation. These are termed “marginal” and “joint” labelling processes respectively, and we show that there are significant differences in their posterior predictive performance. Further investigation of the generalized functional Dirichlet process reveals that a more fundamental difference exists. Whereas marginal labelling models necessarily 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 generalized functional Dirichlet process and the functional Dirichlet process. We find that the explicit modelling of the partitioning leads to more tractable computational and ivPrevious papers introduced a variety of extensions of the Dirichlet process to the functional 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 observation. These are termed “marginal” and “joint” labelling processes respectively, and we show that there are significant differences in their posterior predictive performance. Further investigation of the generalized functional Dirichlet process reveals that a more fundamental difference exists. Whereas marginal labelling models necessarily 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 generalized functional Dirichlet process and the functional Dirichlet process. We find that the explicit modelling of the partitioning leads to more tractable computational and iv more structured posterior predictive behaviour than in the generalized functional Dirichlet process, while still offering increased flexibility over the functional Dirichlet process. Finally, we extend the partitioned functional Dirichlet process to the bivariate case.