Mathematical Sciences: Asymptotic Methods for Bayesian and Likelihood Analysis

这项研究的主要目的是提供广泛的 适用的贝叶斯分析方法基于准确的， 高阶渐近理论 主要的分析工具是 改进和采用的是基于拉普拉斯的方法， 逼近积分和弱收敛理论 分布。 参数化的大小的影响， 将探讨近似误差和诊断工具， 将获得对近似的适当性的评估。 多峰后验密度问题的近似解 隐定义函数的边缘后验密度 将被驱逐。 后验的一阶行为 一般非参数先验分布 密度和危险函数将被调查， 扩展二阶近似到 非参数，无限维的问题将被探讨。 贝叶斯方法旨在提高我们的能力， 将现有（先前）信息纳入决策， 预测策略。 贝叶斯方法仍然相当 直到最近的计算机革命。 当前 这一领域的兴奋和活动源于 最近获得的计算能力带来的 计算机革命和洪水带来的迫切需要 与任何感兴趣的主题相关的信息。 这些 科学家们是致力于大幅改善 贝叶斯方法的计算能力。 他们的研究 将创造更强大的统计技术， 在决策过程中综合利用信息。

The primary goal of this research is to provide widely applicable methodology for Bayesian analysis based on accurate, higher-order asymptotic theory. The main analytical tools to be refined and employed are based on Laplace's method for approximating integrals and the theory of weak convergence of distributions. The effect of the parameterization on the size of apporximation errors will be explored and diagnostic tools for assessing the adequacy of an approximation will be obtained. Approximations for problems with multimodal posterior densities and margianl posterior densities of implicitly defined functions will be dervied. The first order behavior of posterior distributions based on general nonparametric prior distributions on densities and hazard functions will be investigated and the possibility of extending second order approximations to nonparametric, infinite dimensional problems will be explored. Bayesian methods are aimed at improving our ability to incorporate existing (prior) information into decision and prediction strategies. The Baysian approach has remained rather philosophical until the recent computer revolution. The current thrust of excitement and activity in this area derives from the recently obtained computational capability brought by the computer revolution and the critical need brought by inundation of information relevant to any subject of interest. These scientists are leaders in the effort to dramatically improve computational capabilities for Bayesian methods. Their research will create more powerful statistical techniques for comprehensively using informaion in the decision making process.