Efficient Computation in Multi-level Models

多层模型中的有效计算近年来，应用统计学中出现了一种新的趋势-构建特定于应用的模型变得越来越可行，这些模型被设计成考虑到任何特定数据生成机制中固有的结构。长期以来，这种模型一直在理论上被倡导，但最近，用于统计分析的新计算工具(例如，硬件、软件和算法)的发展已经开始将这种模型适合于日常实践。当然，还有很多工作要做。这种方法的灵活性是有代价的-它们需要特定于问题的编码，计算时间很长，并且在确定收敛方面存在困难。这项提议旨在使用新开发的高效蒙特卡罗技术来解决其中一些困难。PIS计划通过为一些重要的模型开发新的算法来研究关于这些有效方法的行为和扩展应用的一些突出的理论问题，这些模型是这些方法的主要候选。投资促进机构参与了几个正在进行的实质性数据分析项目(例如，计算生物学和高能天体物理学)，这些项目既有助于澄清相关的理论问题，也将从新的方法中受益。这项研究的计算目标绝不是目的本身，而是一种改进数据分析和统计推断的手段。正如近年来所清楚地表明的那样，改进的计算工具可以开辟统计应用的全新领域，并增加可靠性，从而改善统计推断。研究将集中在新发展的蒙特卡罗技术，如多点Metropolis以及条件、联合和边际数据增强方法。多点Metropolis推广了Metropolis-Hastings算法，允许在每次迭代中有多个相关方案。因此，多点方法更能跳得更远，不太可能陷入局部模式，因此可以显著改善混合。条件、联合和边际增强方法已经在广泛的模型(例如混合效果模型、有限混合模型、多变量t模型、概率广义线性模型和广义线性混合模型、泊松图像模型等)中显著提高了EM和数据增强算法的性能。特别是，这些新算法保持了EM和DA的稳定收敛特性，有时所需的计算时间减少了99%以上。这些方法，特别是串联的方法，有可能在统计实践中显著改进和扩展马尔可夫链蒙特卡罗。这项计划由数学科学和天文科学部和数学和物理科学局的多学科活动办公室共同资助。

EFFICIENT COMPUTATION IN MULTI-LEVEL MODELSIn recent years, a new trend has been growing in applied statistics---it is becoming ever more feasible to build application specific models which are designed to account for the structure inherent in any particular data generation mechanism. Such models have long been advocated on theoretical grounds, but recently the development of new computational tools (e.g., hardware, software, and algorithms) for statistical analysis has begun to bring such model fitting into routine practice. Of course, much work remains to be done. The flexibility of such methods comes at a cost---they require problem specific coding, long computation times, and present difficulties in ascertaining convergence. This proposal aims to tackle some of these difficulties using newly developed efficient Monte Carlo techniques. The PIs plan to study a number of outstanding theoretical questions concerning the behavior and extended application of these efficient methods by developing new algorithms for a number of important models which are prime candidates for these methods. The PIs are involved in several on-going substantive data analytic projects (e.g., in computational biology and high energy astrophysics) which both help to clarify the relevant theoretical questions and stand to benefit from the new methodology. The computational goals of this research are by no means an end unto themselves, but rather a means to improved data analysis and statistical inference. As has been so clearly illustrated in recent years improved computational tools can open up whole new areas of statistical application, as well as increase reliability, thus improving statistical inference.Research will focus on such newly developed Monte Carlo techniques as multi-point Metropolis and the methods of conditional, joint, and marginal data augmentation. Multi-point Metropolis generalizes the Metropolis-Hastings algorithm by allowing multiple dependent proposals at each iteration. As a consequence the multi-point method is more able to jump further, is less likely to be caught in a local mode, and thus can substantially improve mixing. The methods of conditional, joint, and marginal augmentation have already substantially improved performance of the EM and Data Augmentation algorithms in a wide range of models (e.g., mixed-effects models, finite mixture models, multivariate t-models, probit generalized linear models and generalized linear mixed model, Poisson image models, etc.). In particular, these new algorithms maintain the stable convergence properties of EM and DA while sometimes reducing the required computation time by over 99%. These methods, especially in tandem, have the potential to significantly improve and extend Markov Chain Monte Carlo in statistical practice. This program is being jointly funded by theDivision of Mathematical Sciences and Astronomical Sciences and the Office of MultidisciplinaryActivities from the Directorate of Mathematical and Physical Sciences.