Statistical Inferences and Markov Chains, Admissibility, and Strong Inconsistency

DMS 96-26601伊顿这项研究涉及到对统计推断的评估，这些推断被表示为数据相关的概率分布。特别令人感兴趣的推论包括后验分布和预测分布，它们是由不适当的先验分布和贝叶斯定理的正式应用而得出的。关于推论可采性的问题导致了新的和强大的技术的发展，这些技术基于统计决策理论中的问题和涉及对称马尔可夫链的重现/跨科学问题之间的基本联系。这种基本的联系在这项研究中得到了进一步的发展，同时也进一步发展了推论的强烈不一致的某种程度上的联系。这项研究的应用包括评估和改进高维模型中的推理技术，如大规模环境模型中出现的模型和试图描述全球变化的模型。统计学的许多问题，例如在环境研究中出现的问题，涉及许多变量的同时描述和/或同时预测。在这种情况下，人们经常谈到“高维问题”。本研究涉及与这些高维问题相关的统计方法的构建、评价和比较。最近的发展表明，这项研究中出现的问题，令人惊讶的是，与在一个看似无关的概率领域出现的某些问题密切相关。这种联系有望在涉及高维问题的实践和理论问题上产生令人兴奋的新见解。***

DMS 96-26601 Eaton This research involves the evaluation of statistical inferences which are expressed as data dependent probability distributions. Inferences of particular interest include both posterior distributions and predictive distributions which are derived from improper prior distributions coupled with the formal application of Bayes Theorem. Questions concerning the admissibility of inferences have lead to the development of new and powerful techniques based on a fundamental connection between issues in statistical decision theory and recurrence/transcience issues involving symmetric Markov Chains. This fundamental connection is further developed in this research along with the somewhat allied noton of strong inconsistency for inferences. Applications of this research include the evaluation and improvement of inferential techniques in high dimensional models such as those which arise in large scale environmental models and those attempting to describe global change. %%% Many problems of statistics, such as those which arise in environmental studies, involve the simultaneous description and/or simultaneous prediction of many variables. In such cases, one often speaks of "high dimensional problems". This research involves the construction, evaluation and comparison of statistical methods which are relevant in these high dimensional problems. Recent developments have shown that the problems which arise in this research are, rather surprisingly, closely connected with certain problems which arise in a seemingly unrelated area of probability. This connection promises to yield exciting new insights regarding both practical and theoretical issues involving high dimensional problems. ***