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The recent growth in the availability of biomedical data promises to reshape healthcare by ushering in an era of personalized medicine where data can be used to diagnose and treat patients with pinpoint accuracy. Truly realizing this goal requires building statistical models that individually model patient variations such as age, sex, and genetic makeup, which leads to a combinatorial growth in the number of parameters in a model and noisy estimates. Fortunately, Bayesian hierarchical models, along with recent computational advances, provide a solution to this issue. By naturally embedding the hierarchical structure that many datasets exhibit into the model, these models allow for separate estimates that capture population-level variation and simultaneously avoid noise via regularization to a population mean.In this thesis, we describe novel models and computational methods for Bayesian hierarchical modeling of biomedical data. We begin by describing our contributions to various areas of Bayesian modeling, along with the problems from our applied work that motivated these contributions.Specifically, we describe our disease progression model, which hierarchically models patient disease trajectories. The model, which was motivated by our applied work on Alzheimer's Disease, utilizes I-splines to capture the characteristic monotonic shape of dementia disease trajectories, along with Dirichlet distributions over the coefficients of these I-splines to hierarchically model these trajectories. Next we describe our work on using the Givens Representation of orthogonal matrices to infer models with orthogonal matrix parameters, such as factor models, in a general Bayesian framework. We describe the innovations in our method along with our motivating hierarchical example based on the analysis of protein biomarkers of coagulopathic trauma patients. Next, we describe a mechanistic model of coagulopathy that relates clotting assay data to protein concentrations, effectively providing a fast and convenient way for clinicians to understand key protein markers involved in clotting.Next we transition specifically to Hamiltonian Monte Carlo (HMC) and elucidate the connection between multiscale posterior distributions and the efficiency of HMC. We describe the issue of numerical stability inside HMC and present our implicit HMC algorithm for efficiently sampling non-Gaussian posterior distributions.Lastly, we provide a summary of our contributions along with ideas for future work.

We introduce an approach based on the Givens representation for posterior inference in statistical models with orthogonal matrix parameters, such as factor models and probabilistic principal component analysis (PPCA). We show how the Givens representation can be used to develop practical methods for transforming densities over the Stiefel manifold into densities over subsets of Euclidean space. We show how to deal with issues arising from the topology of the Stiefel manifold and how to inexpensively compute the change-of-measure terms. We introduce an auxiliary parameter approach that limits the impact of topological issues. We provide both analysis of our methods and numerical examples demonstrating the effectiveness of the approach. We also discuss how our Givens representation can be used to define general classes of distributions over the space of orthogonal matrices. We then give demonstrations on several examples showing how the Givens approach performs in practice in comparison with other methods.

© 2015 Pourzanjani et al. Being able to infer one way direct connections in an oscillatory network such as the suprachiastmatic nucleus (SCN) of the mammalian brain using time series data is difficult but crucial to understanding network dynamics. Although techniques have been developed for inferring networks from time series data, there have been no attempts to adapt these techniques to infer directional connections in oscillatory time series, while accurately distinguishing between direct and indirect connections. In this paper an adaptation of Granger Causality is proposed that allows for inference of circadian networks and oscillatory networks in general called Adaptive Frequency Granger Causality (AFGC). Additionally, an extension of this method is proposed to infer networks with large numbers of cells called LASSO AFGC. The method was validated using simulated data from several different networks. For the smaller networks the method was able to identify all one way direct connections without identifying connections that were not present. For larger networks of up to twenty cells the method shows excellent performance in identifying true and false connections; this is quantified by an area-under-the-curve (AUC) 96.88%. We note that this method like other Granger Causality-based methods, is based on the detection of high frequency signals propagating between cell traces. Thus it requires a relatively high sampling rate and a network that can propagate high frequency signals.

Windsurfers appeared in the leisure market during the past two decades and have now been established as one of the most popular water sports. Their design has been dominated by the qualitative feedback from the users of such craft. They have been able to achieve a speed record of about 40 knots at the perfect sailing conditions. It is believed that a more scientific approach in design, using CFD, will result in a craft capable of breaking this speed record.