Ergodicity of Adaptive MCMC and its Applications
Markov chain Monte Carlo algorithms (MCMC) and Adaptive Markov chain Monte Carlo algorithms (AMCMC) are most important methods of approximately sampling from complicated probability distributions and are widely used in statistics, computer science, chemistry, physics, etc. The core problem to use these algorithms is to build up asymptotic theories for them. In this thesis, we show the Central Limit Theorem (CLT) for the uniformly ergodic Markov chain using the regeneration method. We exploit the weakest uniform drift conditions to ensure the ergodicity and WLLN of AMCMC. Further we answer the open problem 21 in Roberts and Rosenthal [48] through constructing a counter example and finding out some stronger condition which indicates the ergodic property of AMCMC. We find that the conditions (a) and (b) in [46] are not sufficient for WLLN holds when the functional is unbounded. We also prove the WLLN for unbounded functions with some stronger conditions. Finally we consider the practical aspects of adaptive MCMC (AMCMC). We try some toy examples to explain that the general adaptive random walk Metropolis is not efficient for sampling from multi-model targets. Therefore we discuss the mixed regional adaptation (MRAPT) on the compact state space and the modified mixed regional adaptation on the general state space in which the regional proposal distributions are optimal and the switches between different models are very efficient. The theoretical proof is to show that the algorithms proposed here fall within the scope of general theorems that are used to validate AMCMC. As an application of our theoretical results, we analyze the real data about the ``Loss of Heterozygosity" (LOH) using MRAPT.
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