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Bayesian Parameter Estimation and Variable Selection for Quantile Regression A thesis submitted for the degree of Doctor of Philosophy

The principal goal of this work is to provide efficient algorithms for implementing the Bayesian approach to quantile regression. There are two major obstacles to overcome in order to achieve this. Firstly, it is necessary to specify a suitable likelihood given that the frequentist approach generally avoids such specifications. Secondly, sampling methods are usually required as analytical expressions for posterior summaries are generally unavailable in closed form regardless of the prior used. The asymmetric Laplace (AL) likelihood is a popular choice and has a direct link to the frequentist procedure of minimising a weighted absolute value loss function that is known to yield the conditional quantile estimates. For any given prior, the Metropolis Hastings algorithm is always available to sample the posterior distribution. However, it requires the specification of a suitable proposal density, limiting it’s potential to be used more widely in applications. It is shown that the Bayesian quantile regression model with the AL likelihood can be converted into a normal regression model conditional on latent parameters. This makes it possible to use a Gibbs sampler on the augmented parameter space and thus avoids the need to choose proposal densities. Using this approach of introducing latent variables allows more complex Bayesian quantile regression models to be treated in much the same way. This is illustrated with examples varying from using robust priors and non parametric regression using splines to allowing model uncertainty in parameter estimation. This work is applied to comparing various measures of smoking and which measure is most suited to predicting low birthweight infants. This thesis also offers a short tutorial on the R functions that are used to produce the analysis. Declaration I declare that this thesis was composed by myself and that the work contained therein is my own, except where explicitly stated otherwise in the text.