Studying, Improving, and Applying Markov chain Monte Carlo methods

马尔可夫链蒙特卡罗(MCMC)方法允许从高度复杂的概率分布(目标分布)生成数据。Metropolis-Hastings(MH)采样器构成了一类重要的MCMC算法，它们以其通用性(它们可以应用于几乎任何感兴趣的概率分布)和易于实现而闻名。生物统计学、计算机科学、物理学、金融学和应用统计学等不同应用领域的研究人员和实践者广泛使用这些方法。 在应用MH采样器时，有必要选择一种首选的建议分布；正态分布因其可及性而成为流行的选择。我们的想法是在每次迭代中从该提议分布中生成一个候选者；然后该候选者被接受为样本的合适的值(根据特定的接受概率)，或者干脆被丢弃。 现有的采样器在应用程序中并不总是高效地执行。为了处理实际中日益复杂的分布和海量数据集，有必要开发新的采样器和/或改进现有的方法。与其他学科积极合作也很重要，以便开发对解决问题有用的工具。 可逆跳跃MCMC(RJ-MCMC)算法是MCMC策略的扩展，因为它们允许从不同维度的目标分布中采样。这些算法显然构成了贝叶斯模型选择的一个很好的工具，其中参数向量的维度通常是不固定的。它们可以用于多变点分析，其中所考虑的模型允许数据集的不同部分服从不同的概率规律。虽然RJ-MCMC在实践中得到了广泛的应用，但在理论上还没有得到研究。因此，从一个维度移动到另一个维度的机制通常是通过反复试验来选择的。该方案的目的之一是为用户提供调谐RJ-MCMC采样器的理论指导。 我的研究的第二个方面，除了从理论上改进现有的方法外，还在于提高采样器的计算效率。多次尝试Metropolis(MTM)策略允许在给定迭代中生成多个候选对象(通过与通常的MH采样器中的一个相反)。这种方法的一个缺点是在每次迭代时都需要生成一个辅助样本，这大大增加了方法的计算强度。一些研究人员通过重新表达问题来消除对辅助样本的需求，但得到的方法并不优于原始的MTM算法。我相信他们的方法可以通过鼓励马尔可夫链内的运动来改进，从而获得比通常的MTM更有效的算法。 MCMC理论中一个有趣的新途径是在函数空间中进行抽样(通过与兴趣密度中的采样点相反)。这些情况可以在天气预报、海洋学(地下水流动)、医学和安全(图像配准)、物理、金融(从一些随机波动模型中采样)等领域找到。标准的MCMC算法在用于从此类无限维问题中获得越来越准确的样本的网格加密下变得任意缓慢。因此，迫切需要设计能够在这种情况下应用的新的MCMC技术。 最后一个目标是有效地应用在生物学等领域开发的方法；具体地说，我打算使用和改进MCMC方法来研究标记了某些有机体、细菌和病毒的大规模有色网络。特别感兴趣的是这样的网络中的各种类型的路径的分布。

Markov chain Monte Carlo (MCMC) methods allow for data generation from highly complex probability distributions (the target distribution). Metropolis-Hastings (MH) samplers form an important class of MCMC algorithms, which are renowed for their versatility (they can be applied to virtually any probability distribution of interest) and ease of implementation. Researchers and practitioners in various fields of applications such as biostatistics, computer science, physics, finance, and applied statistics extensively use such methods. In applying MH samplers, it is necessary to choose a preferred proposal distribution; the normal distribution is a popular choice due to its accessibility. The idea is to generate, at every iteration, a candidate from this proposal distribution; this candidate is then accepted as a suitable value for the sample (according to a certain acceptance probability), or simply discarded. Existing samplers do not always perform efficienctly in applications. In order to deal with increasingly complex distributions and massive datasets arising in practice, it is necessary to develop new samplers and/or improve existing methods. It is also important to collaborate actively with other disciplines so as to develop tools that are useful for their problematics. Reversible-jump MCMC (RJ-MCMC) algorithms constitute an extension of the MCMC strategy, as they allow sampling from target distributions of varying dimensions. These algorithms obviously constitute a great tool in Bayesian model selection, where the dimensionality of the parameter vector is typically not fixed. They can be used in multiple change-point analysis, where the models considered allow different parts of a dataset to obey different probability laws. Although extensively used in practice, the RJ-MCMC has not been studied theoretically. As a result, the mechanism from moving from one dimension to another is usually chosen by trial and error. One goal of this proposal is to provide users with a theoretical guideline for tuning the RJ-MCMC sampler. A second aspect of my research, besides theoretically improving existing methods, is to improve the computational efficiency of samplers. The multiple-try Metropolis (MTM) strategy allows generating multiple candidates in a given iteration (by opposition to only one in the usual MH sampler). A drawback of this method is the necessity of generating an auxiliary sample at every iteration, which significantly increases the computational intensity of the method. Some researchers have eliminated the need for this auxiliary sample by reexpressing the problem, but the resulting method does not outperform the original MTM algorithm. I believe that their method could be improved by encouraging movements within the Markov chain, so as to obtain an algorithm more efficient than the usual MTM. An interesting new avenue in MCMC theory aims at sampling in spaces of functions (by opposition to sampling points from a density of interest). These situations may be found in weather forecasting, oceanography (goundwater flow), medicine and security (image registration), physics, finance (sampling from some stochastic volatility models), etc. Standard MCMC algorithms become arbitrarily slow under the mesh refinement used to obtain increasingly accurate samples from such infinite-dimensional problems. There is thus a crying need for designing new MCMC techniques that can be applied in such contexts. A last goal is to effectively apply the methodology developed in areas such as Biology; specifically I intend to use and refine MCMC methods for investigating large-scale colored networks labeled with certain organisms, bacteria, and viruses. Of particular interest is the distribution of various types of paths in such networks.