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

该项目将重点研究以下六个领域： 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。在 在前两个领域，将特别强调问题 包括测试和模型选择。 贝叶斯方法 这些问题没有得到充分利用，主要是因为 在这些领域的强大或默认的先验分析还没有被 开发 在条件推理中，主要焦点将是 研究贝叶斯答案具有有效 条件频率论解释 查明这类 情况不仅有基础价值，而且 表示采用条件频率论的情况 方法可以是非常有利的。 在贝叶斯计算中， 重点将放在使用马尔可夫链和其他模拟 进行贝叶斯集成的技术。 贝叶斯分析可能是增长最快的 统计分析的方法，因为它的建模能力 甚至分析极其复杂的情况， 容易地允许多个信息源的组合。的 贝叶斯分析的主要局限性一直受到关注， 对建模假设的敏感性，以及 在高维问题中进行贝叶斯计算。 在这个项目中，我们将解决敏感性问题， 开发本质上“稳健”的贝叶斯模型， 创造了强大的方法， 贝叶斯模型 在计算方面，我们将扩展新的 基于模拟的技术， 具有数百或数千个未知模型的情况 参数

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.