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

该项目将专注于六个领域的研究：鲁棒贝叶斯分析、默认先验贝叶斯方法、决策理论中的条件推理、多元估计、层次建模和贝叶斯计算。在前两个领域，将特别强调涉及测试和模型选择的问题。贝叶斯方法对这些问题的利用严重不足，主要是因为这些领域的鲁棒性或默认先验分析尚未开发。在条件推理中，主要重点将是研究贝叶斯答案具有有效条件频率论解释的情况。识别这种情况不仅具有基础价值，而且还表明采用条件频率论方法可能非常有利的情况。在贝叶斯计算中，重点将放在使用马尔可夫链和其他模拟技术来进行贝叶斯积分。贝叶斯分析可能是发展最快的统计分析方法，因为它能够建模和分析极端复杂的情况，而且它很容易允许多个信息源的组合。贝叶斯分析的主要局限性是对建模假设的敏感性，以及在高维问题中执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“鲁棒”的贝叶斯模型，并通过创建从竞争贝叶斯模型中进行选择的强大方法来解决敏感性问题。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.