Mathematical Sciences: Bayesian Statistical Theory and Methodology

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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.