【作者】 董晓燕；

【导师】 赵联文；

【作者基本信息】 西南交通大学 ， 概率论与数理统计， 2008， 硕士

【摘要】 统计问题中,贝叶斯方法在很多方面已经硕果累累,不过在处理非参数方面却仍存在很大的差距,这主要由于在参数空间上寻找有效先验分布是非常困难的,具体到非参数问题就是在给定样本空间取一个概率分布集。基于Ferguson1973年的文章,在非参数问题中,对先验分布有两方面的要求:(1)样本空间中,相对于概率分布空间上的某些适当的(弱)拓扑,先验分布必须有足够大的支撑。这就保证了先验选择的灵活性与广泛性,以便于找到最适合模型的分布函数。(2)在给定先验分布类和观测样本时,后验分布必须易于计算,至少有可行的计算方法。从而保证在实际中的应用价值。然而这两个要求是相悖的,一方的满足必须以牺牲另一方为条件。我们通常的处理方法是通过放宽第一个条件,而将第二个条件设置为共轭类来构造分布类。参看最近几十年的文章,我们可以发现,在处理非参数贝叶斯问题中用到最多的先验分布,都是现已有的几种具体的先验,如Dirichlet过程,Talifree过程,中立过程,Polya树等。由于先验分布的限制,所以贝叶斯方法在处理非参数问题时,受到了阻力。因此,有必要研究在确定非参数问题中是否存在确定先验分布的一般方法或者是在一定限制条件下确定先验分布的一般方法这一基本问题。本文基于Ferguson对先验分布提出的两方面的要求和现已知的先验分布的构造方法,讨论了在可数样本空间和不可数样本空间上的先验分布的一些构造方法及相应先验分布的性质,并且给出了Dirichlet过程先验在估计后验均值方面的应用。本文主要做了以下几方面的工作:1.给出了先验分布在可数样本空间上的构造,通过规范化构造和Stick-breaking构造两种方法进行说明,并说明了构造方法的可行性。2.给出了先验分布在不可数样本空间上的构造,通过Binning构造,增量过程构造,剖分树构造等六种不同的构造方法确定先验分布。3.讨论了几种先验分布的一些性质和重要结论。4.给出了Dirichlet过程先验在估计后验均值方面的应用。更多还原

【Abstract】 The Bayesian approach to statistical problems,though fruitful in many ways,has been rather unsuccessful in treating nonparametric problems.This is due primarily to the difficulty in finding workable prior distributions on the parameter space,which in nonparametric problems is taken to be a set of probability distributions on a given sample space.Based on the paper of Ferguson in 1973, there are two desirable properties of a prior distribution for nonparametric problems:(1) The support of the prior distribution should be large-with respect to some suitable topology on the space of probability distributions on the sample space.This can assure the feasibility and universality of the prior,so we can find the best model for the distribution.(2) Posterior distributions given a sample of observations from the true probability should be manageable analytically.It requires the Posterior distributions have the same forms as the priors,or they are conjugate classes,or they can easily be computed.These properties are antagonistic in the sense that one may be obtained at the expense of the other.We usually broad a class of prior distributions in the sense of (1),for which (2) is realized by given in the sense of conjugate class.Refer to the papers in the past few decades, the prior distributions we used most in treating nonparametric problems are those prior classes, eg: Dirichlet processes, Tailfree processes, neutral processes, Polya tree and so on. The Bayesian approach to statistical problems has been unsuccessful in treating nonparametic problems. This is due primarily to the limitations of prior distribution. It is necessary to consider whether there is a general method of construct prior distribution under some conditions. Based on two desirable properties of a prior distribution for nonparametric problems and some known prior distributions construction, some methods of construct prior distribution on countable sample spaces and uncountable sample spaces are introduced and given an algorithm to estimate the values of posterior means with a Dirichlet process prior.This paper does the work as following:1.Given methods of construction of prior distributions on countablesample spaces, i.e. construction via normalization and construction via stick-breaking2.Given methods of construction of prior distributions on uncount?able sample spaces, i.e. construct prior via binning,via increasing processes,via partitioning tree and so on.3.Discussed some properties and important facts of prior distribu?tion.4.Given an algorithm to estimate the values of posterior means witha Dirichlet process prior.更多还原