Mathematical Sciences: Bayesian Statistical Theory and Methodology

贝叶斯统计理论和方法的研究可分为五类：(1)发展合适的统计模型和方法，用于分析所选数据和经常遇到的其他类型的非随机样本。将特别强调将选择样本中的信息与不受限制的随机样本中的信息进行比较。(2)合并专家意见的问题，包括对专家报告的意见可能与其真实意见有何不同的正式研究，以及确定某一合并规则是否为可受理规则的条件的发展。(3)最优检查与控制问题，其中需要对随机过程安排一系列代价高昂的检查，以便尽可能快地检测到该过程已经失控或进入某个期望的目标区的事件。(4)最优停止问题，在该问题中，决策者团队中的每个成员只收到关于序贯样本中观测数据的部分信息，他们彼此之间不能直接通信，并且他们必须制定出联合停止过程的最优策略。(5)开发适用于各种法律环境的贝叶斯统计方法。这项研究是在贝叶斯统计的一般领域，这是一种统计决策的方法，纳入了在收集与这一问题有关的新数据之前存在的相关知识。这种方法背后的一般理论并不新鲜，但到目前为止几乎没有实际用途。这是因为需要求解的数学方程需要使用哪怕是一点点先验信息，这需要大量的计算能力和聪明的算法，而直到最近，这种计算资源还没有普遍可用。随着所需资源变得更加普遍，这项研究的潜在影响也越来越大。随着针对具体实际例子的开创性贝叶斯方法的开发，正如这里所建议的那样，这种方法有望得到推广，并彻底改变数据用于决策的方式。

The proposed research on Bayesian statistical theory and methods falls into five categories: (1) The development of appropriate statistical models and methodology for the analysis of selected data and other types of non-random samples often encountered. Particular emphasis will be placed on comparing the information in a selection sample with that in an unrestricted random sample. (2) Problems of combining expert opinions, including a formal study of how the opinions that experts report might differ from their true opinions, and the development of conditions for determing whether a given rule of combination is an admissible one. (3) Problems of optimal inspection and control in which it is necessary to schedule a sequence of costly inspections of a stochastic process in order to detect as quickly as possible the event that the process has gone out of control or has entered some desirable target area. (4) Problems of optimal stopping in which each member of a team of decision makers receives only partial information about the observations in a sequential sample, they connot communicate directly with each other, and they must develop an optimal strategy for jointly stopping the process. (5) The development of appropriate Bayesian statistical methods for use in various legal settings. This research is in the general area of Bayesian statistics, an approach to statistical decision making that incorporates the pertinent knowledge that exists prior to the collection of new data bearing on the issue. The general theory behind this approach is not new, but has been of little practical use to date. This is because the mathematical equations that need to be solved to use even a little bit of prior information require large amounts of computational horsepower and clever algorithms, and, until recently, such computational resources have not been generally available. With the required resources becoming more commonly available, the potential impact of this research grows. As pathbreaking Bayesian methods for specific practical examples are developed, as is proposed here, this approach is expected to spread and to revolutionize the way data is used to make decisions.