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.
","abstract_html":"<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>","abstract_has_math":false,"creators":["Hellmayr, Christoph Stefan"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Gelfand, Alan E"],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017","date_published":"2017","updated_at":"2026-07-24T02:06:56Z","subjects":["Statistics","Bayesian","Functional Data","Non-Parametrics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/10161/16373","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Gelfand, Alan E"]},{"key":"dc:creator","label":"Author","values":["Hellmayr, Christoph Stefan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2018-03-20T17:56:56Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2018-03-20T17:56:56Z"]},{"key":"dc:date.issued","label":"Date","values":["2017"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Statistics","Bayesian","Functional Data","Non-Parametrics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10161/16373"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Non-Parametric Priors for Functional Data and Partition Labelling Models"]}]}],"canonical_facts":{"dc:contributor.advisor":["Gelfand, Alan E"],"dc:creator":["Hellmayr, Christoph Stefan"],"dc:date.accessioned":["2018-03-20T17:56:56Z"],"dc:date.available":["2018-03-20T17:56:56Z"],"dc:date.issued":["2017"],"dc:description.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-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.
"],"dc:identifier.uri":["https://hdl.handle.net/10161/16373"],"dc:subject":["Statistics","Bayesian","Functional Data","Non-Parametrics"],"dc:title":["Non-Parametric Priors for Functional Data and Partition Labelling Models"],"dc:type":["Dissertation"]},"updated_at":"2026-07-24T02:06:56Z"}