In the past fifteen years there has been a dramatic increase of interest in Bayesian mixture models. This is because we are now able to apply Markov chain Monte Carlo (MCMC) and sequential Monte Carlo (SMC) methods. This thesis is concerned with Bayesian mixture modelling via such approaches. The following topics are considered. Firstly, an important problem with Bayesian mixture models is that of label switch- ing. This is caused by the nonidentifiability of the components under symmetric priors. Therefore, Monte Carlo estimates of component specific quantities will be identical and thus useless for inference. We review the existing solutions to the label switching problem and develop a probabilistic method to deal with it. Secondly, another difficulty is actually simulating from trans-dimensional measures induced by mixtures. To solve this problem, we present an extension of population-based MCMC. We provide a result related to the uniform ergodicity of a population Markov kernel. We also give an example of a population algorithm for simulating from a Bayesian multivariate mixture model. Thirdly, a problem for SMC samplers (Del Moral et al.; 2005) is that of low particle diversity for multimodal targets. In such situations, even under reasonable Markov kernels, poor Monte Carlo estimates may be derived. We present an interacting SMC method which seeks to maintain a diverse set of particles. We establish some convergence results. Additionally, we show how the methodology may be used for a Bayesian mixture model and, using adaptive methods, a mixture problem in population genetics. Finally, we develop statistical methodology for geochronological data. The main characteristics of geochronological data are heterogeneity and measurement error. The former property means that mixture models are appropriate for its analysis. We establish that current methods for analyzing such data are not always suitable. Therefore, we develop Bayesian mixture models for this task.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:484721 |
| Date | January 2006 |
| Creators | Jasra, Ajay |
| Contributors | Holmes, Chris ; Stephens, David |
| Publisher | Imperial College London |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/10044/1/1257 |
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