Bayesian Strategies for Complex Statistical Models

This thesis aims to address one algorithm based question and one model based problem. The first aim of this thesis is to illustrate the performance of the noisy Metropolis-Hastings algorithm, and to compare to the approximate Bayesian computation (ABC) methods and Markov chain Monte Carlo (MCMC) methods in the context of point process models. Chapter 2 implements the comparisons on two repulsive spatial point process models one of which is a typical doubly-intractable model. The likelihood normalising constant of another one is tractable with some approximation, however it is computationally expensive to evaluate. The second aim is to deal with weighted networks with excessive non-interactions which are treated as zeros. Zero-inflated models is applied to classify those zeros into two different types: missing zeros and true zeros. We interpret a missing zero if no interaction is observed between two individuals while they actually have interacted or potentially interact with each other. Thus the true zeros are those correctly recorded zeros. We embed this zero-inflated framework in the setting of the stochastic block model to facilitate clustering of the nodes in the presence of excessive zero edges. Chapter 3 introduces our zero-inflated Negative-Binomial stochastic block model (ZINB-SBM) which is able to deal with networks with missing zeros, clustering of nodes and over-dispersed weights. We focus on directed networks in this chapter and the number of clusters is fixed for the partially collapsed Gibbs sampler combining with the Metropolis-Hastings step we implement for the inference. A modified version of the integrated classification log-likelihood criterion is also proposed for the model selection. Chapter 4 instead focuses on undirected cases with Poisson distribution embedded inside the zero-inflated stochastic block model (ZI-SBM) scheme in order to explore the efficiency, security and redundancy structures of a real criminal network. Such structures correspond to a missing data imputed version of the ZI-SBM scheme we illustrate in chapter 3. One of the Gibbs-type priors is further proposed for the latent clustering variable, so that the partially collapsed Gibbs sampler, which we use to infer the model, is able to automatically infer the number of clusters.