贝叶斯非参数统计中的先验的估计
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摘要
贝叶斯非参数统计是一个新兴的但发展迅速的统计研究领域,不但其理论成果非常丰富,其实际应用范围也十分广泛。然而,贝叶斯非参数统计的传统研究着眼于一种纯贝叶斯的多层先验结构,其中需要事先确定先验分布。一旦不能事先容易地确定先验,特别是因为贝叶斯非参数统计通常要求一个复杂的过程先验,那么这一多层先验结构将会受到挑战和质疑。传统的贝叶斯非参数统计分析的这一缺陷促使我们采用一种更加灵活,更加稳健的统计框架—经验贝叶斯分析—来实施统计推断和统计建模。这是因为在进行经验贝叶斯分析时,人们通常基于观测数据来估计先验参数,而不是事先主观地给定。另外,众所周知,如果可识别性不成立,那么基于观测值来估计参数将会变得毫无意义,而且,可识别性也是证明参数估计或者后验分布的渐近收敛性质的前提条件之一。许多统计学家试图找出可识别性成立的条件,但据我们所知,确实存在许多关于有限混合可识别性的理论成果；但可数无穷混合的可识别性仍然很少被研究到,因此也是一个开放的问题。例如,Ferguson(1983)指出Dirichlet过程先验的混合模型,作为一个可数无穷混合的特例,其可识别性尚未解决。为了解决贝叶斯非参数统计中这些问题和挑战,基于经验贝叶斯的框架和几种不同的数据结构：一元数据,多元数据和单调缺失数据,我们尝试分别对几类过程先验中的参数进行估计。另外,对可数无穷混合的可识别性问题,我们试图提出某些更加方便验证的充分条件。本博士论文的主要内容如下所述。
首先,在第一章中,我们对贝叶斯非参数统计进行一个全面的回顾,包括：人们为什么使用贝叶斯非参数统计,其简要的历史发展,其丰富的理论成果和实际应用。我们以回顾一系列文献的方式,阐述了贝叶斯非参数统计中的计算问题、未来的研究方向和可能面临的挑战。在此之后,我们引入了人们所熟知的经验贝叶斯假定和几种数据结构。这些数据结构非常普遍且颇具代表性,因而能够表达对多种实际数据进行统计建模的设想。
在第二章,通过引入分布集上的良序和序列的一致收敛,我们提出了一个可数无穷混合可识别性成立的充分条件,并且相信此充分条件比Tallis(1969)所提出的无穷维矩阵条件更加容易验证。然后我们运用此充分条件去重新验证了已知可识别性成立的几个例子,进而考查了几个新分布族的可数无穷混合的可识别性,其中包括：正态分布,伽玛分布,柯西分布,非中心卡方分布和广义逻辑斯蒂分布。
第三章涉及单调缺失数据机制下Dirichlet过程先验中的先验参数估计问题。我们试图基于经验贝叶斯框架下的部分观测数据,来估计DP(α,α)中的未知精度参数α和未知概率测度a。我们发现,在Dirichlet过程先验的假定下,数据的缺失不影响精度参数α的估计,因其可以通过极大化某个似然函数来有效地估计。然而,对假定密度函数存在的概率测度a而言,我们必须借助于处理缺失数据的非参数密度估计方法来对其进行估计。精度参数α的估计的强相合性和渐近正态性在非常一般的条件下得到了证明,同时我们也证明了a的密度估计的L1收敛性。另外基于二维单调缺失数据,通过最小化渐近积分均方误差,我们提出了此密度估计的最优窗宽选取方法,并且发现此密度估计优于单调缺失数据下其他已有的方法。
第四章涉及一元数据下Polya tree先验中的先验参数估计问题,也就是说,在事先确定好分划(?)的情况下,我们试图基于数据来估计PT(π,(?))中参数集合(?)中的参数。首先,我们回顾了Polya tree先验的基本模型和理论性质,然后定义了几类Polya tree先验,并给出了使得它们取绝对连续分布集作为支撑的充分条件。之后,我们提出了Polya tree先验中的先验参数的两种估计：矩估计和极大似然估计,并讨论了相应的理论性质,其中包括该模型与beta-binomial分布之间的联系。最后,我们提供了各种估计的数值模拟来验证各自的理论表现。
在第五章中,基于经验贝叶斯框架下的多元观测数据,我们进行了多元Polya tree先验的参数估计。这一节可以视为上述一元Polya tree先验的参数估计问题的一个多元推广,而且此处的经验贝叶斯分析确实类似于一元情形下的相应的分析。首先我们给出多元Polya tree先验的定义和理论性质,然后提出相应的数据结构和模型假设。接下来,我们给出多元Polya tree先验中的先验参数的矩估计和极大似然估计,并讨论了该模型与Dirichlet-multinomial分布之间的联系。最后,我们进行了数值模拟,并通过相应的图表来说明我们所提出的经验贝叶斯估计的理论性质。
Bayesian nonparametrics is a relatively young, yet fast growing field of statistics, in which it not only produces a large number of theoretical achievements but also widely applies itself in various substantive fields and directions. However, its classical research spots focus on a pure Bayes hierarchical structure, with the priors specified presumably. This framework might be challenged once the prior could not be assigned easily, especially in view of the fact that Bayesian nonparametrics requests a rather complicated process prior. This disadvantage of traditional Bayesian nonparametric analysis stimulates us to employ a more flexible and robust framework—empirical Bayes—to proceed statistical inference and modeling. This is due to in empirical Bayes analysis, the prior parameters are estimated based on observations, rather than pre-specified. On the other hand, as is well-known, estimating parameters via observations would not make sense if without identifiability, and identifiability is one of the preconditions to prove asymptotic conver-gence properties of parameter estimates or posterior distributions. Many statisticians tried to figure out under what conditions identifiability will naturally holds and to our knowledge, there exist plenty of fruits on identifiability of finite mixtures while identifia-bility for countably infinite mixture models is still rarely studied thus an open problem. For example, as a special kind of countably infinite mixture, Ferguson (1983) pointed out the identifiability of Dirichlet process mixture model was yet unsolved and ambiguous. Motivated by these challenges in Bayesian nonparametrics, we try to estimate the prior parameters of several process priors under an empirical Bayes framework with different data structures:univariate data, multivariate data, even monotone missing data. We also try to generate some conveniently-applied sufficient conditions for identifiability of countably infinite mixtures. This dissertation is organized as the following chapters.
Firstly, we provide a comprehensive review on Bayesian nonparametrics in Chapter1, including why we apply Bayesian nonparametrics, a brief history of Bayesian non-parametrics, its abundant theoretical achievements and applications. We also discuss the computational issues, future research directions and potential challenges to Bayesian nonparametrics, via recalling a series of literatures from numerous statisticians. Also, we introduce several types of data structures, accompanied with a familiar empirical Bayes assumption. We believe these data structures are so general and representative that they would recapitulate our efforts on modeling various practical datasets.
In chapter2, it presents a sufficient condition for identifiability of countably infinite mixtures, which is expressed by means of well-ordering on the sets of distributions and uniform convergence of series. We think this sufficient condition is easier to be checked than the infinite-dimensional matrix conditions generated by Tallis (1969). Then this suf-ficiency is applied to revisit some examples for which the identifiability is well-established and further explore the identifiability for several novel distribution families, including normal, gamma, cauchy, noncentral χ2, generalized logistic distributions.
The next chapter is concerned with estimating the prior parameters in Dirichlet pro-cess priors when the data are monotonically missing. Our goal is to estimate the unknown precision parameter a and the probability measure a of DP(α, α), with partially observed data in the empirical Bays framework. We find that, under Dirichlet process priors, data missing has no impact on the estimation of the precision parameter a, which can be effec-tively estimated by maximizing certain likelihood function. For the probability measure a, on the contrary, one has to resort to nonparametric density estimation methodologies for missing data when it assumes a has density. The strong consistency and asymp-totic normality of the estimates of a are given under very weak conditions, then the L1convergence of a's density estimate is also proved. Besides, the optimal selection of the bandwidths via minimizing the asymptotic mean integrated squared errors involved in this density estimate is examined for2-dimensional missing data. We find that our density estimate behaves better than some existing approaches under monotone missing data.
Chapter4is related to the estimation of prior parameters in Polya tree priors with univariate observations. Specifically, we try to obtain the empirical Bayes estimates of prior parameters in (?) of Polya tree prior PT(Ⅱ,(?)) with presumably specified partition II based on data. Firstly, we overview the basic model and theoretical properties of Polya tree priors, then define several kinds of Polya tree priors and give sufficiencies to take absolutely continuous distributions as their supports. Two types of estimates to prior parameters in Polya tree priors—the maximum likelihood estimate and the moment estimate—are given, where associated properties are also illustrated, including the rela-tionship between this model and Beta-binomial distribution. We also present numerical simulations on various estimates to validate their respective theoretical performances.
In the last chapter, we estimate the prior parameters of multivariate Polya tree prior based on multivariate observed samples within an empirical Bayes framework. This part could be regarded as a multivariate generalization to the univariate Polya tree, and the corresponding empirical Bayes analyses indeed are similar to the ones of univariate case. At the commencement, we give the definition and prove several theoretical properties of multivariate Polya tree priors, accompanied with data structure and model assumptions. Then we provide the moment estimate and maximum likelihood estimate to the prior parameters of multivariate Polya tree priors, and discuss the relationship between this model and the Dirichlet-multinomial distribution. Numerical simulations are also given to illustrate the performance of the empirical Bayes estimates, in forms of several tables and graphs.
首先,在第一章中,我们对贝叶斯非参数统计进行一个全面的回顾,包括：人们为什么使用贝叶斯非参数统计,其简要的历史发展,其丰富的理论成果和实际应用。我们以回顾一系列文献的方式,阐述了贝叶斯非参数统计中的计算问题、未来的研究方向和可能面临的挑战。在此之后,我们引入了人们所熟知的经验贝叶斯假定和几种数据结构。这些数据结构非常普遍且颇具代表性,因而能够表达对多种实际数据进行统计建模的设想。
在第二章,通过引入分布集上的良序和序列的一致收敛,我们提出了一个可数无穷混合可识别性成立的充分条件,并且相信此充分条件比Tallis(1969)所提出的无穷维矩阵条件更加容易验证。然后我们运用此充分条件去重新验证了已知可识别性成立的几个例子,进而考查了几个新分布族的可数无穷混合的可识别性,其中包括：正态分布,伽玛分布,柯西分布,非中心卡方分布和广义逻辑斯蒂分布。
第三章涉及单调缺失数据机制下Dirichlet过程先验中的先验参数估计问题。我们试图基于经验贝叶斯框架下的部分观测数据,来估计DP(α,α)中的未知精度参数α和未知概率测度a。我们发现,在Dirichlet过程先验的假定下,数据的缺失不影响精度参数α的估计,因其可以通过极大化某个似然函数来有效地估计。然而,对假定密度函数存在的概率测度a而言,我们必须借助于处理缺失数据的非参数密度估计方法来对其进行估计。精度参数α的估计的强相合性和渐近正态性在非常一般的条件下得到了证明,同时我们也证明了a的密度估计的L1收敛性。另外基于二维单调缺失数据,通过最小化渐近积分均方误差,我们提出了此密度估计的最优窗宽选取方法,并且发现此密度估计优于单调缺失数据下其他已有的方法。
第四章涉及一元数据下Polya tree先验中的先验参数估计问题,也就是说,在事先确定好分划(?)的情况下,我们试图基于数据来估计PT(π,(?))中参数集合(?)中的参数。首先,我们回顾了Polya tree先验的基本模型和理论性质,然后定义了几类Polya tree先验,并给出了使得它们取绝对连续分布集作为支撑的充分条件。之后,我们提出了Polya tree先验中的先验参数的两种估计：矩估计和极大似然估计,并讨论了相应的理论性质,其中包括该模型与beta-binomial分布之间的联系。最后,我们提供了各种估计的数值模拟来验证各自的理论表现。
在第五章中,基于经验贝叶斯框架下的多元观测数据,我们进行了多元Polya tree先验的参数估计。这一节可以视为上述一元Polya tree先验的参数估计问题的一个多元推广,而且此处的经验贝叶斯分析确实类似于一元情形下的相应的分析。首先我们给出多元Polya tree先验的定义和理论性质,然后提出相应的数据结构和模型假设。接下来,我们给出多元Polya tree先验中的先验参数的矩估计和极大似然估计,并讨论了该模型与Dirichlet-multinomial分布之间的联系。最后,我们进行了数值模拟,并通过相应的图表来说明我们所提出的经验贝叶斯估计的理论性质。
Bayesian nonparametrics is a relatively young, yet fast growing field of statistics, in which it not only produces a large number of theoretical achievements but also widely applies itself in various substantive fields and directions. However, its classical research spots focus on a pure Bayes hierarchical structure, with the priors specified presumably. This framework might be challenged once the prior could not be assigned easily, especially in view of the fact that Bayesian nonparametrics requests a rather complicated process prior. This disadvantage of traditional Bayesian nonparametric analysis stimulates us to employ a more flexible and robust framework—empirical Bayes—to proceed statistical inference and modeling. This is due to in empirical Bayes analysis, the prior parameters are estimated based on observations, rather than pre-specified. On the other hand, as is well-known, estimating parameters via observations would not make sense if without identifiability, and identifiability is one of the preconditions to prove asymptotic conver-gence properties of parameter estimates or posterior distributions. Many statisticians tried to figure out under what conditions identifiability will naturally holds and to our knowledge, there exist plenty of fruits on identifiability of finite mixtures while identifia-bility for countably infinite mixture models is still rarely studied thus an open problem. For example, as a special kind of countably infinite mixture, Ferguson (1983) pointed out the identifiability of Dirichlet process mixture model was yet unsolved and ambiguous. Motivated by these challenges in Bayesian nonparametrics, we try to estimate the prior parameters of several process priors under an empirical Bayes framework with different data structures:univariate data, multivariate data, even monotone missing data. We also try to generate some conveniently-applied sufficient conditions for identifiability of countably infinite mixtures. This dissertation is organized as the following chapters.
Firstly, we provide a comprehensive review on Bayesian nonparametrics in Chapter1, including why we apply Bayesian nonparametrics, a brief history of Bayesian non-parametrics, its abundant theoretical achievements and applications. We also discuss the computational issues, future research directions and potential challenges to Bayesian nonparametrics, via recalling a series of literatures from numerous statisticians. Also, we introduce several types of data structures, accompanied with a familiar empirical Bayes assumption. We believe these data structures are so general and representative that they would recapitulate our efforts on modeling various practical datasets.
In chapter2, it presents a sufficient condition for identifiability of countably infinite mixtures, which is expressed by means of well-ordering on the sets of distributions and uniform convergence of series. We think this sufficient condition is easier to be checked than the infinite-dimensional matrix conditions generated by Tallis (1969). Then this suf-ficiency is applied to revisit some examples for which the identifiability is well-established and further explore the identifiability for several novel distribution families, including normal, gamma, cauchy, noncentral χ2, generalized logistic distributions.
The next chapter is concerned with estimating the prior parameters in Dirichlet pro-cess priors when the data are monotonically missing. Our goal is to estimate the unknown precision parameter a and the probability measure a of DP(α, α), with partially observed data in the empirical Bays framework. We find that, under Dirichlet process priors, data missing has no impact on the estimation of the precision parameter a, which can be effec-tively estimated by maximizing certain likelihood function. For the probability measure a, on the contrary, one has to resort to nonparametric density estimation methodologies for missing data when it assumes a has density. The strong consistency and asymp-totic normality of the estimates of a are given under very weak conditions, then the L1convergence of a's density estimate is also proved. Besides, the optimal selection of the bandwidths via minimizing the asymptotic mean integrated squared errors involved in this density estimate is examined for2-dimensional missing data. We find that our density estimate behaves better than some existing approaches under monotone missing data.
Chapter4is related to the estimation of prior parameters in Polya tree priors with univariate observations. Specifically, we try to obtain the empirical Bayes estimates of prior parameters in (?) of Polya tree prior PT(Ⅱ,(?)) with presumably specified partition II based on data. Firstly, we overview the basic model and theoretical properties of Polya tree priors, then define several kinds of Polya tree priors and give sufficiencies to take absolutely continuous distributions as their supports. Two types of estimates to prior parameters in Polya tree priors—the maximum likelihood estimate and the moment estimate—are given, where associated properties are also illustrated, including the rela-tionship between this model and Beta-binomial distribution. We also present numerical simulations on various estimates to validate their respective theoretical performances.
In the last chapter, we estimate the prior parameters of multivariate Polya tree prior based on multivariate observed samples within an empirical Bayes framework. This part could be regarded as a multivariate generalization to the univariate Polya tree, and the corresponding empirical Bayes analyses indeed are similar to the ones of univariate case. At the commencement, we give the definition and prove several theoretical properties of multivariate Polya tree priors, accompanied with data structure and model assumptions. Then we provide the moment estimate and maximum likelihood estimate to the prior parameters of multivariate Polya tree priors, and discuss the relationship between this model and the Dirichlet-multinomial distribution. Numerical simulations are also given to illustrate the performance of the empirical Bayes estimates, in forms of several tables and graphs.
引文
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