Latent space models have proven useful in all sorts of applications involving social network data due to their flexibility and interpretability. This dissertation extends such class of models to other fields where relational data play a fundamental roll.

First, we introduce a novel approach for modeling cognitive social structures (CSSs) data. We rely on a generalized linear model that incorporates a bilinear structure to model transitivity effects within networks, and a hierarchical specification on the bilinear effects to borrow information across networks. The model allows us to perform a formal evaluation of the accuracy of actors' perception about their position in social space, and also to obtain a consensus representation of the social space that accounts for differential perceptual acuity among actors. The performance of the model is evaluated using simulated data as well as two real CSSs reported in the literature.

Next, we study the integration of databases in the context of online social networks (OSNs). We propose a model for discovering multiple profiles of a single user using both profile and network data. Our proposal is able to handle multiple networks and makes straightforward the propagation of uncertainty of record linkage into later analysis. We illustrate our methodology in two different settings, namely, re-identification and identity resolution, using real data from three popular OSNs.

Finally, we study the impact of combining profile and network data in a de-duplication setting. We also assess the influence of a range of prior distributions on the linkage structure, including our proposal. Our proposed prior makes it straightforward to specify prior believes and naturally enforces the microclustering property. Furthermore, we explore stochastic gradient Hamiltonian Monte Carlo methods as a faster alternative to obtain samples for the network parameters. Our methodology is evaluated using the RLdata500 data, which is a popular dataset in the record linkage literature.

Autonomous vehicles are becoming the platform of choice for large-scale exploration of environmental processes, owing to their low cost and dependability of sensors. Standard trajectory planning methods often preplan a trajectory in advance, or are based on information criteria, and do not use the observations taken at earlier locations on the trajectory to decide where the vehicle should go next. In this dissertation, we propose a framework for real-time, adaptive generation of trajectories that are optimal with respect to some exploration goal, with a focus on smooth continuous trajectories for nonholonomic vehicles. We develop an algorithm that enables the vehicle to gather data, reconstruct the environmental process, and generate piecewise optimal trajectory segments that use the data collected at previously visited locations. Our approach is based on Gaussian process priors and Bayesian optimal design. Gaussian processes provide a method to fuse any prior knowledge of the environmental process with the collected data to obtain the most updated estimation of the process. Through Bayesian optimal design, we develop reward functions that explicitly reflect the operational goal and naturally address the exploration-exploitation trade-off in a principled way. We include a number of empirical evaluations of the methodology and show the advantages of our algorithm on different archetypes of spatial processes. Field tests of the planner on an autonomous ground vehicle demonstrate the practical usefulness of our algorithm. We then extend the framework to incorporate supplementary information from different types of off-vehicle sensors. Three incorporation methods, depending on the nature of the supplementary source, are developed and tested, resulting in performance improvement, especially in the early stages of exploration. Finally, we turn to the case of multiple vehicles and develop an extension of our optimal trajectory algorithm that uses collective information to estimate the process and individual trajectory planning for each vehicle. As a result, computation time per vehicle does not increase as more vehicles are added, while the time required to achieve the exploration goal is reduced nearly linearly.

Factor models are widely used across diverse areas of application for purposes that include dimensionality reduction, covariance estimation, and feature engineering. Traditional factor models can be seen as an instance of linear embedding methods that project multivariate observations onto a lower dimensional Euclidean latent space. This thesis discusses a new class of geometric embedding models for multivariate binary and ordinal data in which the embedding space correspond to a spherical manifold, with potentially unknown dimension. The resulting models include traditional factor models as a special case, but provide additional flexibility. Furthermore, unlike other techniques for geometric embedding, the models are easy to interpret, and the uncertainty associated with the latent features can be properly quantified. These advantages are illustrated using both simulation studies and real data on voting records from the U.S. Congress as well as survey applications.

Over the last few years modeling of temporal evolution of network data has become a relevant problem for different applications. In this work, we develop novel statistical methods for modeling time series of directed binary networks. The main interests are identifying events associated with structural changes over time and perform short-term link prediction of the network in future periods.

First, we introduce a Bayesian hidden Markov model that uses a stochastic blockmodel to describe the community structure of the network during each period. This model allow us to monitor structural changes in the network and also perform accurate short-term predictions of future links. As an alternative for link prediction, we propose two multinomial logistic regression models using different lasso type penalties including an extension of the autologistic model for network data. In this setting, we are able to provide fast computational algorithms for estimation and prediction using optimization and full Bayesian inference. The performance of the models is illustrated using both simulated and real data from a financial trading network in the NYMEX natural gas futures market.

There has been a growing interest during recent years in connectomics, which is the study of interconnections or networks within the human brain. This interest has been spurred by the development of new imaging technologies, which allow researchers to peer non-invasively into the human brain and obtain data on connections. Motivated by these datasets, this dissertation develops a novel class of Bayesian regression models which study the relationships between neuro-scientific phenotypes and brain connectome networks of individuals.

First, we introduce a novel approach that develops a regression framework of the brain network (represented in the form of a symmetric matrix) on a continuous phenotypic response. We propose a novel network shrinkage prior on the network predictor coefficient matrix. The proposed framework is able to identify nodes or functional regions in the brain network and interconnections between different regions, significantly related to the phenotypic response. To the best of our knowledge, our framework is the first principled Bayesian framework that enables identification of network nodes and edges significantly related

to the response. The performance of the proposed model is evaluated with respect to a wide range of existing competitors available in the high dimensional frequentist and Bayesian literature using a variety of simulation studies. The proposed model identifies important brain regions and interconnections significantly associated with creativity for a group of subjects.

Next, we extend our model to build network classifiers when a brain connectome network along with a binary response is provided for a group of individuals. Here we develop a broader class of global-local network shrinkage priors which includes the novel prior distribution specified earlier as a special case. We specifically consider two different global-local network shrinkage priors from this class of priors and investigate them using simulation studies. In particular, we assess their performance in terms of network classification and identifying influential network nodes and edges for the purpose of classification. We also demonstrate superior performance of our proposed network classifiers over state-of-the-art high dimensional classification techniques. Another major contribution remains developing theoretical conditions to guarantee asymptotically consistent classification for the proposed framework. In particular, we derive conditions on the number of network nodes, sparsity in the network coefficient matrix as a function of the sample size to achieve asymptotically optimal classification. While theoretical results on high dimensional binary regression with ordinary shrinkage priors have emerged recently, developing theory for our network classifier model involves several additional challenges due to the complex nature of the global local shrinkage prior developed here. The framework is used to classify individuals into high and low IQ groups based on their brain connectomes.

Notably, the work discussed in the last two paragraphs tacitly assumes that all nodes and edges have similar impact on a phenotype for every individual. In our next project, we study a brain connectome data where this assumption is violated. In fact, there is a relatively less developed literature in neuroscience that argues for different groups of individuals having shared relationships between brain networks and phenotypes, though this literature lacks a principled Bayesian approach that takes into account different relationships of nodes and edges with the response for different groups of individuals and facilitates clustering of individuals. Motivated by this problem and our dataset, we have developed a Bayesian network mixture regression model. Simulation studies and analysis of the brain connectome dataset demonstrate superior performance of the proposed approach over the approach described earlier. Simulation studies are also used to evaluate the performance of the proposed approach by varying the true and fitted number of clusters, size of the network and sample size.

For these projects, computationally efficient Bayesian sampling algorithms are developed to enable computations even for reasonably large networks in presence of moderately large sample size.

Hyperspectral imaging is a remote sensing technique widely used in a variety of military and environmental applications. For example, hyperspectral images can be used to detect chemical plumes invisible to the human eye, and to identify their chemical structure. A hyperspectral image is a massive cube of data consisting of thousands of pixels each with dozens of observations over a range of frequencies in the electromagnetic spectrum. Algorithms that use hypothesis testing and assume independence over pixels have shown success in detecting gas clouds, but often fail in identifying chemical components. We approach identification problems in hyperspectral imaging as a variable selection problem, which can be solved robustly by taking advantage of spatial information in the image. For this purpose we develop Bayesian spatial model selection algorithms which use mixtures of g-priors, Gaussian Markov Random Fields, and Gaussian Process Priors to account for correlation among chemicals, to induce spatial dependence among pixels, and to account for nonlinearities in pixel signals. To illustrate the performance of the models we apply them to several partially synthesized hyperspectral images and show that our method outperforms state-of-the-art algorithms, such as the LASSO and Fused LASSO.

The class of Bayesian stochastic blockmodels has become a popular approach for modeling and prediction with relational network data. This is due, in part, to the fact that inference on structural properties of networks follows naturally in this framework. Here, we study the problem of community detection under stochastic blockmodels in different settings.

First, we evaluate a stochastic gradient variational algorithm for stochastic models. Stochastic gradient variational algorithms have become a popular tool for approximate posterior inference in the statistics and machine learning literatures. We develop a new version of the algorithm and compare its performance to that of Markov chain Monte Carlo, the conventional method used to fit Bayesian stochastic blockmodels. We show that, although the SGV algorithm is scalable, its performance can be very poor, specially when there is substantial uncertainty in the community structure in the data.

Then, we turn our attention to the study of multilevel community structures in network data. That is, arrangements in which vertices group to form communities and, in turn, communities group into supercommunities. We propose a Bayesian hierarchical extension of stochastic blockmodel that is capable of identifying and recovering multilevel communities when these are present on the data. Markov chain Monte Carlo as well as variational algorithms are derived and evaluated.

Finally, we introduce a new dynamic stochastic blockmodel that allows us to study the evolution of communities across time. Our approach models both shifts in community membership using a fragmentation-coagulation prior, and changes in the propensities of interaction among communities using a variant of the autoregressive process. Computation is performed using a Markov chain Monte Carlo algorithm.

All models and algorithms are illustrated using both real and simulated data.

Statistical models of price volatility most commonly use low-frequency (daily, weekly, or monthly) returns. However, despite their availability, two types of financial data have not been extensively studied: high-frequency data where sampling periods are on the order of seconds; and open, close, high, and low (OCHL) data which incorporate intraperiod extremes.

The first part of this dissertation focuses on the development of a filtering-based method for the estimation of volatility in high-frequency returns, which contrasts currently popular averaging-based approaches. The second part of this dissertation develops a foundational and novel method for likelihood-based estimation for bivariate OCHL, an approach unfeasible until now.

In Chapter 2, we formulate a discrete-time Bayesian stochastic volatility model for high-frequency stock-market data that directly accounts for microstructure noise, and outline a Markov chain Monte Carlo algorithm for parameter estimation. The methods described in this paper are designed to be coherent across all sampling timescales, with the goal of estimating the latent log-volatility signal from data collected at arbitrarily short sampling periods. In keeping with this goal, we carefully develop a method for eliciting priors. The empirical results derived from both simulated and real data show that directly accounting for microstructure noise in a state-space formulation allows for well-calibrated estimates of the log-volatility process driving prices.

In Chapter 3, we present and motivate the bivariate OCHL problem, enumerate the fundamental limitations of some common out-of-the-box approaches, and present a semidiscrete Galerkin numerical solver for computing the likelihood of the observed data. In addition, we prove the consistency of maximum likelihood estimates under the approximate density given by the solver.

Chapter 4 develops a closed-form, analytic solution to the OCHL likelihood problem in parameter ranges where the Galerkin solver requires near-infinite compute time and memory to produce numerically accurate results. A matching solution is also proposed to interpolate between parameter regions where neither the Galerkin nor analytic solutions are applicable. Thus, we present a method for producing likelihoods based on OCHL data over all model parameter ranges, which is a key requirement in statistical estimation algorithms. We use numerical experiments in both Chapters 3 and 4 to show the validity of our methods and demonstrate the increase in statistical power in estimating price volatility and correlation when using bivariate OCHL data.

Extreme value theory studies the tail behavior of a stochastic process, and plays a key role in a wide range of applications. Understanding and quantifying the behavior of rare events and the associated uncertainties is practically important for risk assessment, since such unexpected events can result in massive losses of wealth and high cost in human life. In this dissertation, we present a Bayesian nonparametric mixture modeling framework for the analysis of extremes with applications in financial industry and environmental sciences. In particular, the modeling is built from the point process approach to analysis of extremes, under which the pairwise observations, comprising the time of excesses and the exceedances over a high threshold, are assumed to arise from a non-homogeneous Poisson process. To relax the time homogeneity restriction, implicit in traditional parametric methods, a nonparametric Dirichlet process mixture model is presented to provide flexibility in estimation of the joint intensity of extremes, the marginal intensity over time, and different types of return level curves for one financial market. This class of models is then expanded to assess the effect of systemic risk in multiple financial markets. In this case, the process generating the extremes is modeled as a superposition of two Poisson process. This approach provides a decomposition of the risk associated with each individual market into two components: a systemic risk component and an idiosyncratic risk component. Finally, we extend the point process framework to model spatio-temporal extremes from environmental processes observed at multiple spatial locations over a certain time interval. Specifically, a spatially varying mixing distribution, assigned a spatial Dirichlet process prior, is incorporated into the model to develop inference for spatial interpolation of risk assessment quantities for high-level exceedances. The modeling approaches are illustrated with a number of simulated and real data examples.