This dissertation develops flexible and computationally efficient Bayesian mixture modeling methods for various types of renewal processes. Renewal processes are temporal point process models whose stochastic mechanism focuses on the times between successive events, or inter-arrival times. They have been applied in a variety of fields, including system reliability, earthquake recurrence modeling, and analysis of neural spike-trains. The homogeneous renewal process assumes that the inter-arrival times are independent and identically distributed, being a generalization of the homogeneous Poisson process where inter-arrival times are exponentially distributed. Various extensions of this basic model have been proposed, of which discrete marks and time-varying hazards are relevant to this work.

We first propose a Bayesian nonparametric mixture modeling framework for homogeneous renewal process densities. Selection of the mixture kernel and prior specification are guided by specific features of renewal processes. The definition of a renewal process requires finite mean for the inter-arrival time distribution. We discuss sufficient conditions to satisfy this constraint. In addition, event clustering behavior is often of interest in analyzing renewal process point patterns. Clustering behavior is assessed through the renewal function, which can be obtained from the Laplace transform of the inter-arrival time density, hence kernels with analytical Laplace transform expressions are preferred. We present model details using the gamma density kernel, requiring only a minor restriction on prior hyperparameters to satisfy the finite mean requirement. Motivated by the application area of earthquake recurrence modeling, we also develop a model for decreasing density shapes using a uniform mixture kernel.

Markov renewal processes are a generalization of the homogeneous case where discrete state information is observed with each event. Transitions from one state to another are governed by a Markov chain, and inter-arrival times arise conditionally from transition-specific distributions. For example, earthquake recurrence characteristics may depend on whether the observed magnitudes exceed certain thresholds. Conventional estimation methods for Markov renewal models treat each transition case independently, which facilitates convenient computation but may ignore underlying structure or similarities between cases. Using as foundation the nonparametric mixture modeling framework developed for homogeneous renewal processes, we propose a novel modeling approach for Markov renewal processes where dependence between transition cases is captured through a dependent nonparametric prior. Our proposed framework contains both the homogeneous renewal process and the conventional Markov renewal process as special limiting cases, allowing the degree and nature of dependence to be studied. This method is particularly useful in earthquake recurrence models, where the additional inferences provided by our model reveal interesting patterns in how earthquake magnitudes affect recurrence times. We explore model properties through simulated data and then compare several models applied to an earthquake dataset from Southern California.

Certain extensions of the homogeneous renewal process, such as the time-varying modulated renewal process, are defined in terms of the inter-arrival hazard rate function rather than the density. In these settings, a flexible model applied directly to the hazard function can be more easily adapted to such extensions. Additionally, prior information in some applications may be more naturally expressed on the hazard scale, which may be difficult to integrate into a density-oriented model. We propose a novel basis representation model for hazard functions, using log-logistic hazard basis functions and a nonparametric prior model for the basis coefficients. The result is a flexible and computationally efficient model for renewal process hazard functions. To demonstrate its tractability as a foundation for renewal process extensions, we formulate a nonparametric model for modulated renewal processes.