Monte Carlo and Quasi-Monte Carlo Methods for Statistics

该项目扩展和改进了蒙特卡罗抽样技术。其主要思想是将源于准蒙特卡罗抽样的思想融入到蒙特卡罗抽样中。特别是，这个项目将准蒙特卡罗抽样嵌入到马尔可夫链蒙特卡罗模拟中，直到最近才知道这一组合是可能的。这种组合可以非常有效，在某些问题上带来超过100倍的方差减少。项目人员正在确定准蒙特卡罗何时会给马尔科夫链蒙特卡罗带来很大的改进，并寻找新的方法来结合这些方法。这个项目正在改进蒙特卡罗抽样的另一个领域是无界函数的积分。有一些版本的随机准蒙特卡罗抽样获得了比原始准蒙特卡罗抽样更好的收敛速度，至少对于行为良好的被积分者是这样。不幸的是，无界被积函数的问题并没有得到很大的改善。这个项目将随机准蒙特卡罗抽样的好处扩展到无界被积，通过改变变量公式来限制被积函数，同时努力防止被积函数变得太尖峰。该项目还在研究加速马尔可夫链蒙特卡罗混合的回火方法，以及蒙特卡罗和准蒙特卡罗思想在生物信息学问题中的应用。蒙特卡罗抽样几乎应用于科学和工程的每一个分支。最简单的是，它包括使用随机数生成器模拟系统，并记录所发生的事情。在实践中，蒙特卡罗方法被用来通过蛮力计算来解决一些很难用数学方法解决的问题。现实世界中的复杂问题可以很容易地引入到模拟中，这往往会使问题变得太难，无法进行精确的数学处理。准蒙特卡罗方法可以用于通过用非常仔细地选择的比随机数更平衡的数字来代替随机数序列来驱动模拟。其结果通常是给定精度级别的极大加速比，或给定计算时间的极大精度提升。该项目将准蒙特卡罗方法推向了迄今为止被认为无法从中受益的模拟领域。这些模拟技术被称为马尔可夫链蒙特卡罗，被用于许多领域，包括材料科学、教育测试数据集分析、生物医学研究、机器人、计算机图形学和营销。该项目还在寻找其他方法来改进蒙特卡罗方法，并具有同样广泛的受益潜力。

This project extends and improves Monte Carlo sampling techniques. The main idea is to incorporate ideas that originated in quasi-Monte Carlo sampling into Monte Carlo sampling. In particular this project embeds quasi-Monte Carlo sampling into Markov chain Monte Carlo simulations, a combination that until recently was not known to be possible. The combination can be very effective, bringing variance reductions of over 100 fold in some problems. The project personnel are identifying when quasi-Monte Carlo brings a large improvement in Markov chain Monte Carlo, as well as finding new ways to combine the methods. Another area in which this project is improving Monte Carlo sampling is in integration of unbounded functions. There are versions of randomized quasi-Monte Carlo sampling that attain a better convergence rate than the original quasi-Monte Carlo sampling, at least for well behaved integrands. Unfortunately, problems with unbounded integrands don't see much improvement. This project extends the benefit of randomized quasi-Monte Carlo sampling to unbounded integrands by applying a change of variable formula to bound the integrand while endeavoring to prevent the integrand from becoming too spiky. This project is also investigating tempering methods for speeding up the mixing of Markov chain Monte Carlo as well as applications of Monte Carlo and quasi-Monte Carlo ideas to problems in bioinformatics.Monte Carlo sampling is used in just about every branch of science and engineering. At its simplest it involves simulating a system using random number generators, and recording what happens. In practice Monte Carlo methods are used to solve by brute force computation some problems that are too hard to do mathematically. Real world complications that can easily be introduced into a simulation often make a problem too hard for exact mathematical treatment. Quasi-Monte Carlo methods can be used to drive a simulation byt replacing the random number sequence by very carefully chosen numbers that are much more balanced than random numbers are. The result is often a tremendous speedup for a given level of accuracy, or a tremendous increase in accuracy for a given computation time. This project pushes quasi-Monte Carlo methods into simulations that had hitherto been thought incapable of benefiting from them. Those simulation techniques, known as Markov chain Monte Carlo, are used in many areas including materials science, analysis of educational testing data sets, biomedical research, robotics, computer graphics, and marketing. This project is also looking at other ways to improve Monte Carlo methods with similarly broad potential for benefit.