Studying, Improving, and Applying Markov chain Monte Carlo methods

研究、改进和应用马尔可夫链蒙特卡罗方法

• 批准号: RGPIN-2014-03931
• 负责人: Bédard, Mylène
• 金额: $ 1.02万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=644947
• 关键词: Studying; Improving; Applying; Markov; chain

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.

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各个应用领域的研究人员和从业人员广泛使用这些方法。**在应用MH样本时，有必要选择一个首选的建议分布；正态分布是一种受欢迎的选择，因为它易于获取。这个想法是，在每次迭代中，从这个提案分布中产生一个候选人；然后将该候选值作为样本的合适值（根据一定的接受概率）接受，或者干脆丢弃。**现有的采样器在应用中并不总是有效地执行。为了处理实践中出现的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以开发对其问题有用的工具也很重要。**可逆跳跃MCMC （RJ-MCMC）算法构成了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。这些算法显然是贝叶斯模型选择的一个很好的工具，在贝叶斯模型选择中，参数向量的维数通常是不固定的。它们可以用于多变化点分析，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。本提案的目标之一是为用户提供调优RJ-MCMC采样器的理论指导。**我的研究的第二个方面，除了在理论上改进现有的方法，是提高采样器的计算效率。多次尝试Metropolis （MTM）策略允许在给定的迭代中生成多个候选项（通过反对通常的MH采样器中只有一个）。该方法的缺点是每次迭代都需要生成一个辅助样本，这大大增加了该方法的计算强度。一些研究人员通过重新表达问题来消除对辅助样本的需要，但所得到的方法并没有优于原始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采样（与从感兴趣的密度中采样点相反）。这些情况可能出现在天气预报、海洋学（地下水流动）、医学和安全（图像配准）、物理学、金融（从一些随机波动模型中抽样）等领域。为了从无限维问题中获得越来越精确的样本，标准的MCMC算法在网格细化下变得任意缓慢。因此，迫切需要设计新的MCMC技术来应用于这种情况。**最后一个目标是有效地应用在生物学等领域开发的方法；具体来说，我打算使用和改进MCMC方法来研究用某些生物体、细菌和病毒标记的大规模彩色网络。特别令人感兴趣的是这种网络中各种类型路径的分布。