Maria De Iorio

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UCL, Dept of Statistical Science


Markov Random Fields at Multiple Resolutions and an ANOVA Model for Dependent Random Measures

The majority of this thesis concerns the analysis of Markov random field (MRF) models. Firstly we describe a method to construct a class of coherent MRFs at different scales, overcoming the problem that the marginal Gaussian MRF is not, in general, a MRF. This is based on the approximation of non-Markov Gaussian fields as Gaussian MRFs and is optimal according to different theoretic notions such as Kullback-Leibler divergence. Moreover, we extend the method to intrinsic autoregressions providing a novel multi-resolution framework. Further, we describe a method of combining information from different scales. Our way of reweighting observations is based on distance, ensuring that the marginal posterior distributions under both fine and coarse representations are ``close''. We then consider computational aspects of Gaussian MRFs and propose an approximation algorithm to calculate the posterior distribution of a MRF and to generate samples from it. The algorithm relies on the fact that this posterior is identical to the posterior of a particular dynamic linear model and turns out to be computationally more efficient, especially for large lattices, than the exact one (Lavine, 1999), without involving any relevant loss of information. In the last part of the thesis we consider dependent non-parametric models for a set of related random probability distributions. We introduce dependence in an ANOVA type fashion in such a way that marginally each random measure follows a Dirichlet Process. We use the dependent Dirichlet process to define ANOVA type dependence across the related random measures and demonstrate the model with data taken from a clinical trial for anti-cancer agents.