Mathematical Sciences: Asymptotic Methods for Bayesian and Likelihood Analysis

本研究的主要目标是基于精确的高阶渐近理论为贝叶斯分析提供广泛适用的方法。要改进和使用的主要分析工具是基于拉普拉斯逼近积分方法和分布的弱收敛理论。将探讨参数化对近似误差大小的影响，并获得评估近似是否适当的诊断工具。对于具有多模态后验密度和隐定义函数的边际后验密度的问题，将推导近似。我们将研究基于密度和危险函数上的一般非参数先验分布的后验分布的一阶行为，并探索将二阶近似扩展到非参数、无限维问题的可能性。贝叶斯方法旨在提高我们将现有（先验）信息纳入决策和预测策略的能力。在最近的计算机革命之前，贝叶斯方法一直是相当哲学的。当前这一领域的兴奋和活跃源于计算机革命所带来的最近获得的计算能力，以及与任何感兴趣的主题相关的信息泛滥所带来的迫切需求。这些科学家是努力显著提高贝叶斯方法计算能力的领导者。他们的研究将创造更强大的统计技术，以便在决策过程中全面使用信息。

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.