Mathematical Sciences: Investigations in Bayesian Analysis,Statistical Decision Theory, and Computation

该项目将侧重于稳健贝叶斯分析、默认先验贝叶斯方法、决策理论中的条件推理、多元估计、分层建模和贝叶斯计算六个领域的研究。在前两个领域，将特别重视涉及测试和型号选择的问题。贝叶斯方法对这些问题的利用严重不足，主要是因为在这些领域还没有开发出稳健的或默认的先前分析。在条件推理中，主要的焦点将是研究贝叶斯答案具有有效的条件频率解释的情况。这种情况的识别不仅在基础上有价值，而且还表明采用条件频率法可能是非常有利的情况。在贝叶斯计算中，重点将是使用马尔可夫链和其他模拟技术来进行贝叶斯积分。贝叶斯分析可能是增长最快的统计分析方法，因为它能够对甚至极其复杂的情况进行建模和分析，而且它很容易结合多种信息源。贝叶斯分析的主要局限性一直是担心对建模假设的敏感性，以及在高维问题中进行贝叶斯计算的困难。在这个项目中，我们将通过开发本质上“健壮”的贝叶斯模型来解决敏感性问题，并通过创建从竞争对手的贝叶斯模型中进行选择的强大方法来解决。在计算方面，我们将扩展基于模拟的新技术，这些技术有望处理具有数百或数千个未知模型参数的情况。

The project will focus on research in the six areas of Robust Bayesian Analysis, Default Prior Bayesian Methodology, Conditional Inference in Decision Theory, Multivariate Estimation, Hierarchical Modelling, and Bayesian Computation. In the first two areas, special emphasis will be paid to problems involving testing and model selection. Bayesian approaches to these problems are severely underutilized, primarily because robust or default prior analyses in these areas have not been developed. In Conditional Inference, the primary focus will be the study of situations in which Bayesian answers have a valid conditional frequentist interpretation. Identification of such situations is not only valuable foundationally, but also indicates situations in which adopting a conditional frequentist approach can be highly advantageous. In Bayesian Computation, emphasis will be on use of Markov Chain and other simulation techniques for carrying out Bayesian integrations. Bayesian analysis is perhaps the most rapidly growing approach to statistical analysis, because of its ability to model and analyze even extremely complex situations, and because it readily allows combination of multiple information sources. The chief limitations of Bayesian analysis have been concern about sensitivity to modelling assumptions, and the difficulty in carrying out Bayesian computations in high dimensional problems. In this project we will attack the sensitivity problem by developing Bayesian models that are inherently "robust," and by creating powerful methods for selecting from among competing Bayesian models. On the computational side, we will extend new techniques based on simulation that have the promise of handling situations with hundreds or thousands of unknown model parameters.