| Summary: | The massive increases in computational power that have occurred over the last two decades have contributed to the increasing prevalence of Bayesian reasoning in statistics. The often intractable integrals required as part of the Bayesian approach to inference can be approximated or estimated using intensive sampling or optimisation routines. This has extended the realm of applications beyond simple models for which fully analytic solutions are possible. Latent variable models are ideally suited to this approach as it provides a principled method for resolving one of the more difficult issues associated with this class of models, the question of the appropriate number of latent variables. This thesis explores the use of latent variable models in a number of different settings employing Bayesian methods for inference. The first strand of this research focusses on the use of a latent variable model to perform simultaneous clustering and latent structure analysis of multivariate data. In this setting the latent variables are of key interest providing information on the number of sub-populations within a heterogeneous data set and also the differences in latent structure that define them. In the second strand latent variable models are used as a tool to study relational or network data. The analysis of this type of data, which describes the interconnections between different entities or nodes, is complicated due to the dependencies between nodes induced by these connections. The conditional independence assumptions of the latent variable framework provide a means of taking these dependencies into account, the nodes are independent conditioned on an associated latent variable. This allows us to perform model based clustering of a network making inference on the number of clusters. Finally the latent variable representation of the network, which captures the structure of the network in a different form, can be studied as part of a latent variable framework for detecting differences between networks. Approximation schemes are required as part of the Bayesian approach to model estimation. The two methods that are considered in this thesis are stochastic Markov chain Monte Carlo methods and deterministic variational approximations. Where possible these are extended to incorporate model selection over the number of latent variables and a comparison, the first of its kind in this setting, of their relative performance in unsupervised model selection for a range of different settings is presented. The findings of the study help to ascertain in which settings one method may be preferred to the other.
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