Adaptive Markov chain Monte Carlo and Copula Dependence Models

该提案有两个主要主题。 马尔可夫链蒙特卡罗(MCMC)技术已经成为统计学家解决广泛科学领域中复杂分析的重要工具，例如卫生研究、遗传学、社会研究和工程学。一些最广泛使用的MCMC算法的成功实现涉及调整过渡核参数，这一过程通常是乏味的、令人沮丧的，并且在具有高维支持的分布的情况下，太难完成。自适应MCMC是自动调整算法的最新和很有前途的发展。这项工作将解决在将自适应MCMC应用于复杂目标时遇到的理论和方法挑战，例如具有高维支持的多峰分布。开发的算法将在计算发挥核心作用的一些统计学领域产生立竿见影的影响，包括非线性回归模型或集群算法及其在遗传学和健康研究中的应用。 现代统计分析的核心是对随机变量之间的相关性进行建模和理解。Copula提供了一种灵活的替代方法，允许对边际分布和联合相依结构的模型进行单独的规范。条件Copulas的现代发展使我们能够调整协变量的相关性结构，并导致更广泛的可能应用，包括线性和非线性回归模型。我建议研究并进一步发展条件Copula模型的贝叶斯方法。提出的方法的优点包括对边际估计继承的误差进行现实的评估，灵活的Copula选择方法，以及能够实施在这种情况下尚未探索的模型平均想法。

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.