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In many areas of the world, seismic hazards pose a great risk to both human and natural populations. In particular, earthquake-induced tsunamis are especially dangerous to many areas in the Pacific. The study and quantification of these seismic events can both help scientists better understand how these natural hazards occur and help at-risk populations make better preparations for these events. However, many events of interest occurred too long ago to be recorded by modern instruments, so data on these earthquakes are sparse and unreliable. To remedy this, a Bayesian method for reconstructing the source earthquakes for these historical tsunamis based on anecdotal data, called TsunamiBayes, has been developed and used to study historical events that occurred in 1852 and 1820. One drawback of this method is the computational cost to reconstruct posterior distributions on tsunami source parameters. In this work, we improve on the TsunamiBayes method by introducing higher-order MCMC methods, specifically the Hamiltonian Monte Carlo (HMC) method to increase sample acceptance rate and therefore reduce computation time. Unfortunately the exact gradient for this problem is not available, and so we make use of a surrogate gradient via a neural network fitted to the forward model. We examine the effects of this surrogate gradient HMC sampling method on the posterior distribution for an 1852 event in the Banda Sea, compare results to previous results collected usisng random walk, and note the benefits of the surrogate gradient in this context.

Along with being one of the most populated regions of the world, Indonesia has one of the highest natural disaster rates worldwide. One such natural disaster that Indonesia is particularly prone to are tsunamis. Tsunamis are primarily caused by earthquakes, volcanoes, landslides and debris flows. To effectively allocate resources and create emergency plans we need an understanding of the risk factors of the region. Understanding the source events of destructive tsunamis of the past are critical to understanding the these risk factors. We expand upon previous work focusing on earthquake-generated tsunamis to consider landslide-generated tsunamis. Using Bayesian inference and modern scientific computing we construct a posterior distribution of potential landslide sources based on anecdotal data of historically observed tsunamis. After collecting 30,000 samples we find a landslide source event provides a reasonable match to our anecdotal accounts. However, viable landslides may be on the edge of what is physically possible. Future work creating a coupled landslide-earthquake model may account for the weaknesses with having a solely landslide or earthquake source event.

In many environmental and climate applications, extreme data are spatial by nature, and hence statistics of spatial extremes is currently an important and active area of research dedicated to developing innovative and flexible statistical models that determine the location, intensity, and magnitude of extreme events. In particular, the development of flexible sub-asymptotic models is in trend due to their flexibility in modeling spatial high threshold exceedances in larger spatial dimensions and with little or no effects on the choice of threshold, which is complicated with classical extreme value processes, such as Pareto processes. In this thesis, we develop new flexible sub-asymptotic extreme value models for modeling spatial and spatio-temporal extremes that are combined with carefully designed gradient-based Markov chain Monte Carlo (MCMC) sampling schemes and that can be exploited to address important scientific questions related to risk assessment in a wide range of environmental applications. The methodological developments are centered around two distinct themes, namely (i) sub-asymptotic Bayesian models for extremes; and (ii) flexible marked point process models with sub-asymptotic marks. In the first part, we develop several types of new flexible models for light-tailed and heavy-tailed data, which extend a hierarchical representation of the classical generalized Pareto (GP) limit for threshold exceedances. Spatial dependence is modeled through latent processes. We study the theoretical properties of our new methodology and demonstrate it by simulation and applications to precipitation extremes in both Germany and Spain. In the second part, we construct new marked point process models, where interest mostly lies in the extremes of the mark distribution. Our proposed joint models exploit intrinsic CAR priors to capture the spatial effects in landslide counts and sizes, while the mark distribution is assumed to take various parametric forms. We demonstrate that having a sub-asymptotic distribution for landslide sizes provides extra flexibility to accurately capture small to large and especially extreme, devastating landslides.

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The Brigham Young University Women's Volleyball Team recorded and rated all skills (pass, set, attack, etc.) and recorded rally outcomes (point for BYU, rally continues, point for opponent) for the entire 2006 home volleyball season. Only sequences of events occurring on BYU's side of the net were considered. Events followed one of these general patterns: serve-outcome, pass-set-attack-outcome, or block-dig-set-attack-outcome. These sequences of events were assumed to be first-order Markov chains where the quality of each contact depended only explicitly on the quality of the previous contact but not on contacts further removed in the sequence. We represented these sequences in an extensive matrix of transition probabilities where the elements of the matrix were the probabilities of moving from one state to another. The count matrix consisted of the number of times play moved from one transition state to another during the season. Data in the count matrix were assumed to have a multinomial distribution. A Dirichlet prior was formulated for each row of the count matrix, so posterior estimates of the transition probabilities were then available using Gibbs sampling. The different paths in the transition probability matrix were followed through the possible sequences of events at each step of the MCMC process to compute the posterior probability density that a perfect pass results in a point, a perfect set results in a point, and so forth. These posterior probability densities are used to address questions about skill performance in BYU women's volleyball.

Clinical drug trials are costly and time-consuming. Bayesian methods alleviate the inefficiencies in the testing process while providing user-friendly probabilistic inference and predictions from the sampled posterior distributions, saving resources, time, and money. We propose a dynamic linear model to estimate the mean response at each dose level, borrowing strength across dose levels. Our model permits nonmonotonicity of the dose-response relationship, facilitating precise modeling of a wider array of dose-response relationships (including the possibility of toxicity). In addition, we incorporate an adaptive approach to the design of the clinical trial, which allows for interim decisions and assignment to doses based on dose-response uncertainty and dose efficacy. The interim decisions we consider are stopping early for success and stopping early for futility, allowing for patient and time savings in the drug development process. These methods complement current clinical trial design research.

Marine turtle conservation is most successful when it is based on sound data incorporating life history, historical population stability, and gene flow among populations. This research attempts to provide that information through two studies. In chapter I, I identify historical patterns of gene flow, population sizes, and contraction/expansion during major climatic shifts. In chapter II, I reveal a life history characteristic of loggerhead turtles previously undocumented. I identify a pattern of juvenile recruitment to foraging grounds proximal to their natal nesting beach. This pattern results in a predictable recruitment pattern from juvenile foraging ground aggregations to local rookeries. This research will provide crucial information to conservation managers by demonstrating how sensitive marine turtles are to global climate change. In the second component of my research, I demonstrate how threats posed to juvenile foraging grounds will have measurable effects on rookeries proximal to those foraging grounds. The addition of this basic life history information will have dramatic effects on marine turtle conservation in the future, and will serve as the basis for more thorough, forward-looking recovery plans.

The statistical analysis of cointegration is crucial for inferring shared stochastic trends between variables and is an important area of Econometrics for analyzing long-term equilibriums in the economy. Bayesian inference of cointegration involves the identification of cointegrating vectors that are determined up to arbitrary linear combinations, for which the Gibbs sampler is often used to simulate draws from the posterior distribution. However, economic theory may not suggest linear relations and regime-switching models can be used to account for non-linearity. Modeling cointegration and regime-switching as well as the combination of them are associated with highly parameterized models that can prove to be difficult for Markov Chain Monte Carlo techniques such as the Gibbs sampler. Hamiltonian Monte Carlo, which aims at efficiently exploring the posterior distribution, may thus facilitate these difficulties. Furthermore, posterior distributions with highly varying curvature in their geometries can be adequately monitored by Hamiltonian Monte Carlo. The aim of the thesis is to analyze how Hamiltonian Monte Carlo performs in simulating draws from the posterior distributions of models accounting for cointegration and regime-switching. The results suggest that while it is not necessarily the case that regime-switching will be identified, Hamiltonian Monte Carlo performs well in exploring the posterior distribution. However, high rates of divergences from the true Hamiltonian trajectory reduce the algorithm to a Random Walk to some extent, limiting the efficiency of the sampling.

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