【作者】 李勇；

【导师】 李裕奇；

【作者基本信息】 西南交通大学 ， 应用数学， 2006， 硕士

【摘要】 先验分布的确定问题是贝叶斯统计首要的基本问题。这一问题涉及两个方面:一是如何利用一些先验信息或无信息来确定有关参数的先验分布的问题:二是对于同一参数,已有不少的先验分布可供选择,如何选择出一个恰当的合理先验的问题,即先验分布的选择问题。这两个问题实质上是先验分布确定问题的两个方面。 对于第一个问题,贝叶斯统计学家们已经取得了很多的方法。本文根据统计推断所利用的三种信息(先验、总体、样本信息)的不同应用,尝试着对这些常用的方法进行了一定的整理,并提炼了一个概念——数据控制下的先验,以期区别无信息先验和非主观先验概念。 对于先验的选择问题,本文基于这样的一个基本观点:在可选先验类中选择一个合理先验的问题,与在参数空间中估计一个恰当的参数作为模型的参数的问题类似。再借助于(经典和贝叶斯)统计学推断的理论和方法,得到了先验分布选择的方法: 首先,考虑了在经典统计推断原则下先验分布的选择。证明了ML-Ⅱ先验就是本文的似然合理先验。 其次,仅用先验信息考虑了先验的选择。证明了多层贝叶斯先验就是本文的(先验)均值合理先验。这表明确定先验的多层先验法也可以作为选择合理先验的方法,这是对多层先验法一个新的应用。 最后,考虑了基于贝叶斯分析的先验选择。先给出求解先验的后验分布计算方法和参数的后验分布计算方法,再根据相应的后验分布,得到先验选择的相应方法。同时也证明了在先验信息为均匀分布时,本文的贝叶斯似然合理先验和后验似然合理先验就是ML-Ⅱ先验。并举例阐述了方法的应用。更多还原

【Abstract】 The establishment of the prior distribution is most important for Bayesian statistics,which contains two aspects , one is how to establish the prior distribution of parameters by a bit of non-information or prior information, the other is how to choose a proper prior distribution from so many ones. A criterion of classifying for these methods for the first problem is proposed in this paper which has been discussed by many Bayesian statisticalists. According different applications to the information we collect and category these methods which are used usually and introduce a new notion—prior under the control of the dates in order to clarify the conceptions between non-informative prior and non-subjective prior.A basic idea is expressed for choosing prior distribution, which is same to the choice of parameter for the model in the space of parameters. A set of theory for this is built in virtue of classical statistics in this paper and a verification for that is given. And then the ways of prior selection which utilizes prior information only is proposed. The verification for that hierarchical Bayesian prior is a kind of reasonable prior means indicate that the method establishing hierarchical Bayesian prior of prior is a proper method to chose prior. This is a new application to hierarchical Bayesian prior.At last, methods to choose prior resort Bayesian analysis are given from two aspects. Numerations of posterior distribution to gain prior distribution and parameter is given respectively, the corresponding method to choose prior is educed consequently. At the same time we conclude that Bayesian likelihood reasonable prior and posterior likelihood reasonable prior are uniform with classical ML--II prior when the distribution of prior is equal distribution.更多还原