Adaptive Markov chain Monte Carlo and Copula Dependence Models

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The proposal has two main themes.Markov chain Monte Carlo (MCMC) techniques have become an important tool in the statistician's arsenal for solving complex analyses in a wide range of scientific areas, e.g. health studies, genetics, social studies and engineering. A successful implementation of some of the most widely used MCMC algorithms involves tuning the transition kernel parameters, a process that is often tedious, frustrating and, in the case of distributions with support in high dimensions, too difficult to complete. Adaptive MCMC is a recent and promising development that automatizes the tuning of the algorithm. This work will tackle theoretical and methodological challenges encountered when applying Adaptive MCMC to complex targets such as multimodal distributions with high dimensional support. The algorithms developed will have immediate impact in a number of areas in statistics where computation plays a central role, including nonlinear regression models or clustering algorithms and their applications in genetics and health studies.Central to modern statistical analysis are the modeling and understanding of the dependence between random variables. Copulas provide a flexible alternative that allows separate specifications of the models for the marginal distributions and the joint dependence structure. The modern development of conditional copulas allows us to adjust the dependence structure for covariates and leads to a much wider range of possible applications, including linear and nonlinear regression models. I propose to study and further develop Bayesian methods for conditional copula models. The advantages of the proposed methods include a realistic assessment of the error inherited from the marginal estimation, flexible copula selection methodology and the ability to implement model averaging ideas which have not been explored in this setting.