Output Analysis for Markov Chain Monte Carlo

与传统的蒙特卡罗方法相比，马尔可夫链蒙特卡罗方法（MCMC）有几个缺点。特别是，每个从业者在实现MCMC时面临的一个重要问题是何时停止计算。通常情况下，需要结合经验和特别的方法来做出这个决定。因此，人们不得不怀疑推理的质量。研究者研究了顺序固定宽度方法，该方法允许为感兴趣的数量构造一个区间估计量。区间估计量描述了点估计的置信度。研究者利用这一点来研究当MCMC计算旨在估计目标分布的一般数量时有效停止规则的发展。这些方法要求马尔可夫链以几何速率收敛，这反过来意味着在感兴趣的设置中存在点估计的极限分布。因此，研究者研究了两大类贝叶斯模型中遇到的马尔可夫链的收敛率。MCMC方法已经成为应用统计学家（以及其他学科的许多科学家）工具箱中的标准技术。事实上，说它彻底改变了应用统计学，尤其是贝叶斯统计学，一点也不为过。不幸的是，MCMC方法并不总是被谨慎地使用，导致文献中可疑的说法。特别是，很少有人努力在推论结论中包括不确定性的度量。通过为统计学家和其他科学家提供使用MCMC在他们的研究环境中进行推理的有效技术，严格解决这些不确定性措施中的停止规则问题，可以增强研究和教育的基础设施。

Markov chain Monte Carlo (MCMC) has several disadvantages when compared to classical Monte Carlo methods. In particular, an important issue that every practitioner faces when implementing MCMC is when to stop the computation. Typically, a mixture of experience and ad hoc methods is employed to make this decision. Thus one is forced to wonder about the quality of the inference. The investigator studies sequential fixed-width methods that allow construction of an interval estimator for the quantity of interest. The interval estimator describes the confidence in the point estimate. The investigator uses this to study the development of valid stopping rules when the MCMC computation is aimed at estimating general quantities of the target distribution.These methods require the Markov chain to converge at a geometric rate which in turn implies there is a limiting distribution of the point estimate in the settings of interest. Thus the investigator studies the convergence rates of Markov chains encountered in two broad classes of Bayesian models.MCMC methods have become a standard technique in the toolbox of applied statisticians (and many scientists in other disciplines). Indeed, it is not much of an overstatement to say that it has revolutionized applied statistics, especially that of the Bayesian variety. Unfortunately, MCMC methods are not always used carefully leading to dubious claims in the literature. In particular, there has been little effort to include measures of uncertainty in inferential conclusions. Rigorously addressing the issue of stopping rules in terms of these measures of uncertainty enhances infrastructure for research and education by providing statisticians and other scientists valid techniques for using MCMC to make inference in their research setting.