Bayesian methods for system reliability and community detection

Bayesian methods are valuable for their natural incorporation of prior information and their practical convenience for modeling and estimation. This dissertation develops flexible Bayesian parametric methods for system reliability and Bayesian nonparametric models for community detection. The Bayesian parametric models proposed allow the assessment of system reliability for multi-component systems simultaneously. We start with a model that considers lifetime data at every component. Then we generalize to a unified framework with heterogeneous information. We demonstrate this unified methodology with pass/fail, lifetime, and degradation data at both the system level and the component level. Further, we propose a Bayesian melding approach to combine prior information from multiple levels. For community detection, we propose a series of statistical models based on Bayesian nonparametric techniques. These statistical models provide a natural approach for identifying communities in networks using only data on edges. We take advantage of the Bayesian nonparametric approach to include an important feature in our models: the number of communities is an implied parameter of the model, which is therefore inferred during estimation. We also introduce an “Erdős Rényi” group for those nodes that do not belong to communities. Other important aspects of this series of models include increasing flexibility of modeling probabilities for edge presence, linking these probabilities to community sizes, and obtaining communities from posterior samples under a decision theory framework. When presenting our models, we discuss model selection and model checking, which are necessary considerations when applying statistical approaches to real problems

A local structure graph model for network analysis

The statistical analysis of networks is a popular research topic with ever widening applications. In this work, we introduce a new class of models for network analysis, called local structure graph models (LSGMs). The approach specifies a network model through local features and allows for an interpretable and controllable local dependence structure. In particular, LSGMs are formulated by a set of full conditional distributions for each network edge, e.g., the probability of edge presence/absence, which depend functionally on neighborhoods or subcollections of other network edges. Hence, LSGMs correspond to a type of Markov Random Field (MRF) model applied to graph edges. The modeling features and interpretation of LSGMs are demonstrated through several numerical studies and illustrated through a network data example involving tornado occurrences. LSGMs are also shown to provide an alternate specification of another popular class of models for random graphs, belonging to exponential random graph models (ERGMs), which specify a model through a joint distribution on the entire collection of graph edges. An ERGM induces conditional distributions and neighborhoods, rather than explicitly defining them as in the LSGM approach. As one consequence of its conditional specification, LSGMs have the advantage of allowing direct control and separate interpretation of parameters influencing large-scale (e.g., marginal means) and small-scale (i.e., dependence) structures in a graph model. This is possible with LSGMs through so-called centered parameterizations of MRF models, which ERGMs are shown to lack. The centered parameterization and conditional specification of LSGMs further provide important advantages in graph modeling when incorporating covariate information from nodes, as illustrated with two further network data examples. However, the centered parameterization was developed for MRFs under an assumption of pairwise-only dependence, meaning that dependence is modeled between pairs of dependent edges only. This particular dependence structure may be inappropriate for modeling network data that exhibit transitivity or a prevalence of triangles within the network, which has been identified as an important feature of various networks. Consequently, the centered parameterization for MRFs is extended to account for triples of dependent edges in LSGMs. This extension then allows for the explicit modeling of transitivity in LSGMs, while retaining the same interpretable separation and control of large- and small-scale effects in a graph model and facilitating the use of covariate information. At the same time, the ability to model transitivity does not imply that this model feature should be commonly used or applied without cautious model diagnostics, which are currently lacking for graph models and for ERGMs in particular. By developing simulation-based model assessments for random graphs, we provide in-depth examinations and analyses of two commonly-used example networks, demonstrating that real network data may not, in fact, support the inclusion of transitivity in a graph model

Bayesian methods for system reliability and community detection

Bayesian methods are valuable for their natural incorporation of prior information and their practical convenience for modeling and estimation. This dissertation develops flexible Bayesian parametric methods for system reliability and Bayesian nonparametric models for community detection. The Bayesian parametric models proposed allow the assessment of system reliability for multi-component systems simultaneously. We start with a model that considers lifetime data at every component. Then we generalize to a unified framework with heterogeneous information. We demonstrate this unified methodology with pass/fail, lifetime, and degradation data at both the system level and the component level. Further, we propose a Bayesian melding approach to combine prior information from multiple levels. For community detection, we propose a series of statistical models based on Bayesian nonparametric techniques. These statistical models provide a natural approach for identifying communities in networks using only data on edges. We take advantage of the Bayesian nonparametric approach to include an important feature in our models: the number of communities is an implied parameter of the model, which is therefore inferred during estimation. We also introduce an “Erdős Rényi” group for those nodes that do not belong to communities. Other important aspects of this series of models include increasing flexibility of modeling probabilities for edge presence, linking these probabilities to community sizes, and obtaining communities from posterior samples under a decision theory framework. When presenting our models, we discuss model selection and model checking, which are necessary considerations when applying statistical approaches to real problems.</p