Bayesian Optimal Sequential Design for Random Function Estimation

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。在这个项目中，研究者开发了用于随机函数估计的贝叶斯最优序贯设计的计算方法。随机函数估计，无论是在贝叶斯非参数回归的背景下，还是在空间/时空过程的分析中，都已经成为大多数科学领域中普遍使用的工具。虽然估计随机函数的方法已经相当成熟，但对这类问题的最优设计还处于初级阶段。这项建议提出了一个研究计划，为随机函数估计的贝叶斯最优序贯设计开发了一个新的计算框架。该计算框架基于进化马尔可夫链蒙特卡罗(EMCMC)，它结合了遗传或进化算法的思想和马尔可夫链蒙特卡罗的能力。这个框架能够考虑观测的一般模型，如广义线性模型和法线的尺度混合。此外，这种方法很容易在基于基函数的回归函数空间上容纳多个协变量和一般先验，例如基于样条和高斯核的回归函数。随机函数的估计出现在许多应用领域，例如环境科学、流行病学、气候学和工程学。工程中一个重要的例子是计算机模型输出的统计近似，例如流体流动模拟器和火箭助推器模拟器的近似。通常，科学家希望在许多不同的实验条件下运行这样的模拟器。然而，通常情况下，每次运行计算机模型都非常昂贵和耗时。处理这些资源约束的有效方法是在相对较少的实验条件下运行模拟器，并用统计非参数模型来近似模拟器的输出。所提出的计算方法允许以顺序的方式选择贝叶斯最优实验条件，即根据从先前实验条件中学到的知识来选择下一个实验条件。因此，建议的设计点顺序选择方法将在成本和效率方面产生巨大改进。贝叶斯方法的另一个重要应用是时空环境过程的动态监测。统计设计问题是确定监测网的几个站点的位置。在一些监测站是移动的情况下，所提出的方法导致了一个最优的自适应监测网络，该网络在控制成本的同时最大化学习。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). In this project, the investigator develops computational methods for Bayesian optimal sequential design for the estimation of random functions. Random function estimation, either in the context of Bayesian nonparametric regression or in the analysis of spatial/spatio-temporal processes, has become an ubiquitous tool in most areas of science. While methods for the estimation of random functions are reasonably well developed, optimal design for such problems is in its infancy. This proposal presents a research program that develops a novel computational framework for Bayesian optimal sequential design for random function estimation. This computational framework is based on evolutionary Markov chain Monte Carlo (EMCMC), which combines ideas of genetic or evolutionary algorithms with the power of Markov chain Monte Carlo. This framework is able to consider general models for the observations, such as generalized linear models and scale mixtures of normals. In addition, this methodology easily accommodates multiple covariates and general priors on the space of regression functions based on basis functions such as splines and Gaussian kernels. Finally, this framework allows optimality criteria with general utility functions that may include competing objectives, such as for example minimization of costs, minimization of the distance between true and estimated functions, and minimization of the prediction error.Estimation of random functions arises in many application areas, such as for example environmental science, epidemiology, climatology, and engineering.An important example in engineering is the statistical approximation of computer model output, e.g., approximation of fluid flow simulators and rocket booster simulators. Usually, scientists want to run such simulators for many different experimental conditions. However, typically each run of a computer model is extremely expensive and time consuming. An effective way to deal with these resource constraints is to run the simulator for a relatively small number of experimental conditions and to fit a statistical nonparametric model to approximate the output of the simulator. The proposed computational methods allow Bayesian optimal choice of experimental conditions in a sequential fashion, that is, the next experimental conditions are chosen based on what has been learned from the previous experimental conditions. Thus, the proposed methodology for sequential choice of design points will result in huge improvements in cost and efficiency. Another important application of the proposed Bayesian methodology is in dynamic monitoring of spatio-temporal environmental processes. The statistical design problem is to decide where to locate the several stations of a monitoring network. In case some of the monitoring stations are mobile, the proposed methodology leads to an optimally adaptive monitoring network which keeps costs under control while maximizes learning.