Estimation and Inference via Computational Statistics Algorithms

为了使用复杂的统计模型准确地分析大型数据集，统计计算是必不可少的。所使用的算法必须高效、准确和可靠，以便分析有效和有用，并导致高质量的统计推断和对关键未知量的估计。我计划用我的NSERC研究经费来调查计算统计算法，特别是马尔可夫链蒙特卡罗（MCMC）算法，从各种角度。我的大部分工作将涉及算法的理论基础，使用数学概率论分析它们的收敛性、性能和改进。我还计划研究不同的算法设计和改进。此外，我将尝试将这些算法应用于来自各种主题的数据集。在所有情况下，我将重点关注这些算法的属性、性能和应用。我计划研究的一些具体的方法和理论问题包括：*如何改进流行的模式合并模拟和并行回火MCMC算法，使高窄模式在高温下不会被忽略？*当Metropolis算法应用于比以前论文中研究的特殊情况更一般的目标分布时，其最佳缩放率和接受率是多少？*由于某些现代MCMC应用程序需要近似计算，当MCMC算法的估计精度受到轻微“扰动”时，它会受到怎样的影响？*当参数数量和数据量趋于无穷大时，“模型选择”MCMC算法的计算复杂度是如何增长的？* MCMC算法如何更好地“适应”，以提高它们在飞行中的性能，同时仍然收敛到正确的数量？我还计划将计算统计算法应用于各种大规模的真实数据集，包括：*癌症治疗患者数据：哪种药物治疗对哪种类型的患者最有效，可以发现哪些隐藏模式？*学生成绩数据：哪些因素影响学生的专业选择和未来的成功？*森林生长数据：是否可以将加拿大森林中树木种群样本的“地面真实”测量结果与公开的卫星图像进行比较，以校准卫星图像以用于未来的树木估计？

Statistical computation is essential in order to analyse large data sets accurately using complicated statistical models. The algorithms used have to be efficient and accurate and reliable in order for the analysis to be valid and useful, and lead to high-quality statistical inference and estimates of key unknown quantities. I plan to use my NSERC research grant to investigate computational statistics algorithms, especially Markov chain Monte Carlo (MCMC) algorithms, from a variety of perspectives. Much of my work will concern the theoretical foundations of the algorithms, analysing their convergence and performance and improvements using mathematical probability theory. I also plan to study different algorithm designs and improvements. In addition, I will try to apply these algorithms to data sets from a variety of subjects. In all cases, I will focus on the properties, performance, and application of these algorithms. Some specific methodological and theoretical questions which I plan to investigate include: * How can the popular mode-merging simulated and parallel tempering MCMC algorithms be improved so that tall narrow modes are not ignored at high temperatures? * What is the optimal scaling and acceptance rate of Metropolis algorithms when applied to target distributions which are much more general than the special cases studied in previous papers? * How is the estimation accuracy of a MCMC algorithm affected when it is slightly "perturbed", due to the approximate computations required of certain modern MCMC applications? * How does the computational complexity of "model-selection" MCMC algorithms grow as the number of parameters and amount of data go to infinity? * How can MCMC algorithms be better "adapted", to improve their performance on the fly, while still converging to the correct quantities? I also plan to apply computational statistics algorithm to various large-scale real data sets, including: * Cancer treatment patient data: What hidden patterns can be found concerning which medical treatments work best for which sorts of patients? * Student grade data: What factors influence students' choice of subject major and future success? * Forest growth data: Can "ground truth" measurements for samples of tree populations from Canadian forests be compared to publicly-available images from satellites, to calibrate the satellite images for future tree estimates?