Efficient Computation in Multi-level Models

近年来，应用统计学中出现了一种新的趋势——建立特定于应用的模型变得越来越可行，这些模型旨在解释任何特定数据生成机制中固有的结构。这种模型长期以来一直在理论基础上被提倡，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拟合带入日常实践。当然，还有许多工作要做。这种方法的灵活性是有代价的——它们需要特定问题的编码、较长的计算时间，并且在确定收敛性方面存在困难。本提案旨在利用新开发的高效蒙特卡罗技术解决其中的一些困难。pi计划通过开发一些重要模型的新算法来研究一些关于这些有效方法的行为和扩展应用的突出理论问题，这些模型是这些方法的主要候选者。pi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，又能从新方法中受益。本研究的计算目标本身并不是目的，而是改进数据分析和统计推断的一种手段。正如近年来所清楚说明的那样，改进的计算工具可以开辟统计应用的全新领域，并提高可靠性，从而改进统计推断。研究将集中在新开发的蒙特卡罗技术，如多点Metropolis和条件、联合和边缘数据增强方法。多点Metropolis通过在每次迭代中允许多个依赖的提案来推广Metropolis- hastings算法。因此，多点方法更能跳得更远，不太可能陷入局部模式，从而可以大大改善混合。条件增强、联合增强和边际增强的方法已经大大提高了EM和数据增强算法在各种模型中的性能（例如，混合效应模型、有限混合模型、多元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）。特别是，这些新算法保持了EM和DA的稳定收敛特性，有时可以将所需的计算时间减少99%以上。这些方法，特别是串联在一起，在统计实践中具有显著改进和扩展马尔可夫链蒙特卡罗的潜力。该项目由数学科学与天文科学司和数学与物理科学局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.