Collaborative Research: Developing a Theoretical and Methodological Framework for High Dimensional Markov Chain Monte Carlo

研究人员研究多变量方法来评估质量，并确保马尔可夫链蒙特卡罗（MCMC）实验的可靠性。 这项工作的强烈动机的贝叶斯方法的功能性神经成像实验的研究，但将适用于任何MCMC模拟。 通常，马尔可夫链输出用于估计包含多个均值和方差参数沿着分位数的参数向量。 一个基本问题是何时终止这样的模拟。 研究人员研究了顺序固定体积停止规则，该规则允许构建用于估计目标向量的置信区域，该置信区域描述了所得估计的可靠性。 使用这些方法要求马尔可夫链以几何速率收敛，这反过来又产生了一个具有相关协方差矩阵的蒙特卡罗误差的极限分布。 估计这个矩阵形成了一个主要组成部分的研究-一个长期悬而未决的问题MCMC输出分析。 研究人员改进了现有的方法，使有效的估计的情况下，目标向量是适度的大。此外，研究人员研究了几种方法来处理真正高维设置中的设置，即当马尔可夫链中的参数比迭代多得多时。 研究人员还正式研究了在功能性神经成像设置中经常遇到的组件式MCMC采样器的收敛速度。 复杂概率模型通常用于帮助理解包括科学、工程、医学、教育和法律在内的一系列领域中的现象。 一个激发研究人员工作的例子是应用认知科学家模拟大脑活动。 这种概率模型的推论通常是通过计算近似得到的。 对于一个广泛使用的计算技术，研究人员研究的收敛性，并制定正式的停止规则，专注于高维实际相关的设置。 这里开发的统计方法将为科学家提供复杂的输出分析技术，从而提高其计算结果的置信度和可靠性。

The investigators study multivariate methods for assessing the quality and ensuring the reliability of a Markov chain Monte Carlo (MCMC) experiment. This work is strongly motivated by research in Bayesian methods for functional neuroimaging experiments, but will be applicable in any MCMC simulation. Usually, Markov chain output is used to estimate a vector of parameters that contains multiple mean and variance parameters along with quantiles. A fundamental question is when to terminate such a simulation. The investigators study sequential fixed-volume stopping rules that allow construction of confidence regions for estimating the target vector, which describe the reliability of the resulting estimates. Using these methods requires that the Markov chain converges at a geometric rate, which in turn yields a limiting distribution for the Monte Carlo error with an associated covariance matrix. Estimating this matrix forms a major component of the research-a long standing open question in MCMC output analysis. The investigators improve on existing methods, which enable effective estimation in the case where the target vector is moderately large. Moreover, the investigators study several methods for handling the setting in truly high-dimensional settings, i.e. when there are many more parameters than iterations in the Markov chain. The investigators also formally study the convergence rates of component-wise MCMC samplers often encountered in the functional neuroimaging settings. Complex probability models are commonly used to help gain understanding of phenomenon in a range of fields including science, engineering, medicine, education, and law. An example that motivates the investigators work is that of applied cognitive scientists modeling brain activity. Inference from such probability models is usually obtained from computational approximations. For a widely used computational technique, the investigators study the convergence properties and develop formal stopping rules focusing on high-dimensional practically relevant settings. The statistical methodology developed here will provide scientists with sophisticated output analysis techniques, leading to greater confidence and reliability for their computational results.