This work provides a Bayesian nonparametric modeling framework for spatial point processes to account for the irregular domain over which the resulting point pattern occurs in the model formulation while balancing flexible inference with efficient implementation. We start with models for the spatial Poisson process, which assumes independence among points given the number of occurrences, and progress to models for Hawkes processes over space and space-time that capture the self-triggering behaviors and relax the independence assumption. We develop nonparametric Bayesian modeling approaches for Poisson processes using weighted combinations of structured beta densities to represent the point process intensity function. For a regular spatial domain, i.e., the unit square, the model construction implies a Bernstein-Dirichlet prior for the Poisson process density, which supports flexible inference about point process functionals with theoretical guarantees. The key contribution is two classes of flexible and computationally efficient models for spatial Poisson process intensities over irregular domains. We address the choice or estimation of the number of beta basis densities and develop methods for prior specification. For the spatial Hawkes process, we develop a semi-parametric modeling approach, leveraging its clustering representation defined as the superposition of an immigrant Poisson process and several offspring Poisson clustering processes centered on parent points generated by earlier generations. We apply the model for the Poisson process developed earlier to the latent immigrant Poisson process and complete the hierarchical model for the spatial Hawkes process with parametric formulations for the offspring Poisson processes and a model for the latent branching structure that specifies lineage among points. Finally, we develop a nonparametric model for the spatial offspring Poisson process under the assumption of spatial isotropy, which reduces modeling for the spatial offspring density to that for the spatial offspring-parent distance density. Such construction allows the model to be free from the implied tail behavior constraints imposed by existing parametric options for the offspring density kernel. We incorporate such a method to model for space-time Hawkes processes. For all methods developed in the dissertation, we design posterior simulation algorithms for full inference on key point process functionals and model checking techniques to examine the model fit. Model capacity is demonstrated with numerous simulation studies, and we focus on real data examples using crime point patterns from the city of Boston.

Traditional approaches to ordinal regression rely on strong parametric assumptions for the regression function and/or the underlying response distribution. While they simplify inference, restrictions such as normality and linearity are inappropriate for most settings, and the need for flexible, nonlinear models which relax common distributional assumptions is clear. Through the use of Bayesian nonparametric modeling techniques, nonstandard features of regression relationships may be obtained if the data suggest them to be present. We introduce a general framework for multivariate ordinal regression, which is not restricted by linearity or additivity assumptions in the covariate effects. In particular, we assume the ordinal responses arise from latent continuous random variables through discretization, and model the latent response-covariate distribution using a Dirichlet process mixture of multivariate normals. We begin with the binary regression setting, both due to its prominent role in the literature and because it requires more specialized model development under our framework. In particular, we use a square-root-free Cholesky decomposition of the normal kernel covariance matrix, which facilitates model identifiability while allowing for appropriate dependence structure. Moreover, this model structure has the computational advantage of simplifying the implementation of Markov Chain Monte Carlo posterior simulation. Next, we develop modeling and inference methods for ordinal regression, including the underdeveloped setting that involves multivariate ordinal responses. Standard parametric models for ordinal regression suffer from computational challenges arising from identifiability constraints and parameter estimation, whereas due to the flexible nature of the nonparametric model, we overcome these difficulties. The modeling approach is further developed to handle ordinal regressions which are indexed in discrete-time, through use of a dependent Dirichlet process prior, which estimates the unique regression relationship at each time point in a flexible way while incorporating dependence across time. We consider several examples involving synthetic data to study the scope of the proposed methodology with respect to inference and prediction under both standard and more complex scenarios for the underlying data generating mechanism. Moreover, a variety of real data examples are used to illustrate our methods. As this methodology is especially well-suited to problems in ecology and population dynamics, we target applications in these areas. In particular, our methods are used to provide a detailed analysis of a data set on rockfish maturity and body characteristics collected across different years.

We propose a Bayesian nonparametric modeling and inference framework for Hawkes processes. The objective is to increase the inferential scope for this practically important class of point processes by exploring flexible models for its conditional intensity function.

As a building block for conditional intensity models, we develop a prior probability model for temporal Poisson process intensities through structured mixtures of Erlang densities with common a scale parameter, mixing on the integer shape parameters. The mixture weights are constructed through increments of a cumulative intensity function modeled nonparametrically with a gamma process prior. This model specification provides a novel extension of Erlang mixtures for density estimation to the intensity estimation setting.

Turning to the main dissertation component, we develop different types of nonparametric prior models for the Hawkes process immigrant intensity and for the excitation function (or its normalized version, the offspring density), the two functions that define the point process conditional intensity. The prior models are carefully constructed such that, along with the Hawkes process branching structure, they enable efficient handling of the complex likelihood normalizing terms in implementation of inference. The methodology is further elaborated to construct a flexible and computationally efficient model for marked Hawkes processes. The motivating application involves earthquake data modeling, where the mark is given by the earthquake magnitude. The proposed model builds from a prior for the excitation function that allows flexible shapes for mark-dependent offspring densities. In the context of our motivating application, the modeling approach enables estimation of aftershock densities that can vary with the magnitude of the main shock, unlike existing marked Hawkes process models for earthquake occurrences.

For all proposed models, we develop approaches to prior specification and design posterior simulation algorithms to obtain inference for different point process functionals. The modeling approaches are studied empirically using several synthetic and real data examples, including data on earthquake occurrences from Japan and from the southwestern US.

The focus of this work is to develop a Bayesian framework to combine information from multiple parts of the response distribution characterized with different quantiles. The goal is to obtain a synthesized estimate of the covariate effects on the response variable as well as to identify the more influential predictors. This framework naturally relates to the traditional quantile regression, which studies the relationship between the covariates and the conditional quantile of the response variable and serves as an attractive alternative to the more widely used mean regression methods. We achieve the objectives through constructing a Bayesian mixture model using quantile regressions as the mixture components.

The first stage of the research involves the development of a parametric family of distributions to provide the mixture kernel for the Bayesian quantile mixture regression. We derive a new family of error distributions for model-based quantile regression called generalized asymmetric Laplace distribution, which is constructed through a structured mixture of normal distributions. The construction enables fixing specific percentiles of the distribution while, at the same time, allowing for varying mode, skewness and tail behavior. This family provides a practically important extension of the asymmetric Laplace distribution, which is the standard error distribution for parametric quantile regression. We develop a Bayesian formulation for the proposed quantile regression model, including conditional lasso regularized quantile regression based on a hierarchical Laplace prior for the regression coefficients, and a Tobit quantile regression model.

Next, we develop the main framework to model the conditional distribution of the response with a weighted mixture of quantile regression components. We specify a common regression coefficient vector for all components to synthesize information from multiple parts of the response distribution, each modeled with one quantile regression component. The goal is to obtain a combined estimate of the predictive effect of each covariate. We consider the following two choices of kernel densities for the mixture model. When the probability of the quantile in each regression component is known, we model the components with the generalized asymmetric Laplace distribution, as its shape parameter introduces flexibility in shape and skewness to the kernel; else when the quantile probabilities are unknown, we use the asymmetric Laplace distribution as kernel density and view its skewness parameter, which is also the quantile probability of the component, as a random quantity and estimate it from the data. Under each kernel density, we formulate the hierarchical structure of the mixture weights and develop the approach to the posterior inference. We consider both parametric and nonparametric priors for the framework, and explore inferences for the number of components to be included. We demonstrate the performance of the method in identification of influential variables with simulation examples and illustrate the posterior predictive inferences in a realty price data from the Boston metropolitan area.

Finally, we extend the framework to apply the methods to specific problems in survival analysis and epidemiology. Both applications involve analyses of two cohorts, which oftentimes exhibit differing responses given the same predictor input. We adapt the proposed framework to model the survival data with right-censoring. For applications in epidemiology, we study the ordering properties of the mixture kernels and incorporate stochastic ordering in the two-cohort mixture framework through structured priors, which conforms with the assumption in certain circumstances of receiver operating characteristic curve estimation. With the adapted models, we carry out cohort-specific identification of influential variables and gain insights into the contribution in estimation and prediction from different parts of the response distribution, which are depicted by the corresponding quantile regression components. We illustrate the applications with a time-to-event data set on length of stay at nursing home and two disease diagnosis data sets, one on adolescent depression and the other on cattle epidemics.

In survival analysis interest lies in modeling data that describe the time to a particular event. Informative functions, namely the hazard function and mean residual life function, can be obtained from the model's distribution function. We focus on the mean residual life function which provides the expected remaining life given that the subject has survived (i.e., is event-free) up to a particular time. This function is of direct interest in reliability, medical, and actuarial fields. In addition to its practical interpretation, the mean residual life function characterizes the survival distribution. In terms of mean residual life function inference, there are two shortcomings present in the current literature. First off, the shape of the functional is often restricted, which forces the researcher to make an assumption that may not be appropriate. Secondly, in cases where the shape of the functional is not parametrically specified, full inference is not obtained. The aim of our research is to provide a modeling approach that yields full inference for the mean residual life function, and is not restrictive on the shape of the functional. In particular, we develop general Bayesian nonparametric modeling methods for inference for mean residual life functions built from a mixture model for the associated survival distribution. Although the prior model is not placed on the mean residual life function directly, our methods offer rich inference for the desired functional. We place a Dirichlet process mixture model on the survival function, and discuss the importance of careful kernel selection to ensure desirable properties for the mean residual life function. We advocate for a mixture model with a gamma kernel and dependent baseline distribution for the Dirichlet process prior. We extend our model to the regression setting by modeling the joint distribution for the survival response and random covariates. This approach provides a flexible method for obtaining inference for the regression functionals when the number of random covariates is small to moderate. We further extend our methods to the scenario where interest lies in comparison of survival between two experimental groups. Typically, we expect the range of survival in the two groups to be the same, but exhibiting different characteristics over that range. Here, we develop a dependent Dirichlet process prior for the mixing distributions having shared locations across the two groups and varying weights to incorporate dependency between populations and achieve richer inferential results. The final scenario we consider is the case in which the researcher believes two populations have ordered mean residual life functions. For such applications, a prior model that incorporates an ordering constraint on the mean residual life functions is attractive. We introduce a mixture of Erlang distributions with weights constructed using Dirichlet process priors that provides the mean residual life ordering result. We demonstrate the utility of our modeling methods through simulation and real data examples. In addition, we draw comparisons with both parametric and semiparametric models.

Scientific discoveries are advanced by flexible and efficient statistical models. Grounded on Bayesian nonparametric modeling techniques, this thesis provides a toolbox for ordinal regression. The toolbox comprises models tailored for various settings, with shared characteristics of flexibility and efficiency. A key building block of the proposed models is a sequential mechanism to treat the ordinal response. Such mechanism implies a factorization of the response distribution that allows efficient, scalable computation through (partial) parallel sampling regarding the response categories. For problems under a cross-sectional setting, we develop nonparametric mixture models, leveraging the same sequential structure to define covariate-dependent mixture weights. Even though covariates are incorporated via linear functions, the mixture models admit flexible ordinal regression relationships, and they relax parametric assumptions for the response distribution. Moving towards modeling the dynamic evolution of ordinal responses from longitudinal studies, the critical insight is to treat the subjects measurements as stochastic process realizations at the corresponding time points. We propose a hierarchical framework that models the mean and covariance structure of the processes nonparametrically and simultaneously, a useful byproduct being a practical method for making predictions on any time scale. For all proposed models, we craft read- ily implementable Markov chain Monte Carlo algorithms that avoid specialized updates or tuning steps. A variety of synthetic and real data examples are used to illustrate the methods. In particular, the models for cross-sectional ordinal regression, along with their extensions, are examined in the context of risk assessment for developmental toxicity studies. We also present a case study in evolutionary biology, in which our method for longitudinal ordinal responses is adapted to identify the impact of temperature on transgenerational responses, using repeated measurements on fish maturation data.