Bayesian nonparametric methods develop priors over a large class of functions that essentially allow any continuous function to be modeled. Though these methods are flexible, they are black box approaches that do not explicitly incorporate additional information on the shape of the curve. In many contexts, though the exact parametric form of the curve is unknown, additional scientific information is available in the form of differential operators. This dissertation develops nonparametric priors over function spaces that are specified by differential operators. Here two novel approaches to nonparametric function estimation are considered. In the first approach the prior is specified by a linear differential equation. The Mechanistic Hierarchical Gaussian process defines a prior over functions consistent with a differential operator. The method is applied to muscle force tracings in a functional ANOVA context, and is shown to adequately describe the between subject variability often seen in such tracings. In the second case a novel spline based approach is considered. Here prior information is specifies the maximum number of extrema (changepoints) for an arbitrary function located on an open set in R. The Local Extrema (LX) spline models the first derivative of the curve and puts a prior over the maximum number of changepoints. This method is applied to animal toxicology studies, human health surveys, and seasonal data; and it is shown to remove artifactual bumps common to other nonparametric methods. It is further shown to superior in terms of estimated squared error loss in simulation studies.Doctor of Philosoph
