Adaptive Markov Chain Monte Carlo methods

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。马尔可夫链蒙特卡罗（MCMC）是一种灵活的计算技术，已被证明对许多科学学科非常有用，是当前贝叶斯推理实现的支柱。最近的研究进展表明，使用自适应方法可以使蒙特卡洛算法更加有效。这项研究计划有两个主要组成部分。第一部分将致力于发展自适应MCMC算法的极限理论。 更具体地说，PI将开发基于预解的鞅逼近技术，以研究各种自适应MCMC算法的中心极限定理和渐近方差估计。本研究活动的第二部分将开发一种新的MCMC算法，用于具有难以处理的归一化常数的概率模型的贝叶斯分析，这是一个目前存在重大计算挑战的主题。这部分的研究是由计算生物学中的蛋白质设计问题所驱动的，但同样的问题也经常出现在许多其他的统计模型中，包括马尔可夫随机场，马尔可夫点过程，马尔可夫链蒙特卡罗是一种成熟的概率分布抽样蒙特卡罗技术.该方法被广泛用于解决许多应用领域中的实质性问题。它在高维数据的情况下特别有用，这在科学和工程研究以及应用中越来越常见。因此，该项目的研究进展可能对这些领域的统计推断方法产生重大影响。通过其理论组成部分，本研究项目将推进普遍的理解，并加强自适应蒙特卡罗方法在实践中的使用。在发展渐近理论的过程中，PI将发展一些成熟的概率工具的原始扩展。该研究还提出了一种新的算法来解决Monte Carlo模拟中最具挑战性的当前问题之一;从统计模型的后验分布中采样，具有难以处理的归一化常数。

This award is funded under the American Recovery and Reinvestment Act of2009 (Public Law 111-5).Markov Chain Monte Carlo (MCMC) is a flexible computational technique thathas proven very useful to many scientific disciplines and is the backboneof current implementations of Bayesian inference. Recent researchdevelopments suggest that the use of adaptive methods can make Monte Carloalgorithms considerably more effective. This research proposal has twomajor components. The first part will contribute to the development of alimit theory for adaptive MCMC algorithms. More Specifically, the PI willdevelop a resolvent-based martingale approximation technique to investigatethe central limit theorem and the asymptotic variance estimation forvarious adaptive MCMC algorithms. The second part of this research activitywill develop a new MCMC algorithm for the Bayesian analysis of statisticalmodels with intractable normalizing constants, a topic that currently posesmajor computational challenges. This part of the research is driven by theprotein design problem in computational biology but the same problem alsofrequently occurs in many other statistical models including Markov randomfields, Markov point processes.Markov Chain Monte Carlo is a well-established Monte Carlo technique forsampling probability distributions. The method is used widely to solvesubstantive problems in many areas of applications. It is especially usefulin situations with high-dimensional data, which is increasingly common inscience and engineering research as well as applications. Thus, theresearch developments from this project can have a significant impact onmethods for doing statistical inference in these areas. Through itstheoretical component, this research project will advance the generalunderstanding and strengthen the use of adaptive Monte Carlo methods inpractice. In the course of developing the asymptotic theory, the PI willdevelop original extensions to some well-established probabilistic tools.The research is also proposing a new algorithm to tackle one of the mostchallenging current problem in Monte Carlo simulation; sampling from theposterior distribution of statistical models with intractable normalizingconstants.