Mathematical Sciences: Sequential Imputations and Gibbs Sampling: Combinations, Comparisons, and Applications

刘9404344由Kong，Liu和Wong(1994)提出的序贯填补方法(以下简称SI)是处理缺失数据问题的一种特殊的多重填补方法，在为遗传连锁分析中的一些困难问题(如Kong et)提供有效的计算方法方面显示出潜力。艾尔1993年；欧文、考克斯和孔令辉1993年)。当相应的完整数据(贝叶斯)预测分布很容易时，该方法通常适用，并且当数据被顺序收集时尤其有用。Gibbs采样器是最近流行的用于从贝叶斯后验分布中进行采样以便于推断的工具。提出的研究目标是比较两种方法的相对优劣，并在一些应用中将两种方法结合起来。在第一个方案中，将SI方法应用于解决数字通信中的盲解卷积问题，其中假设观测信号是离散输入信号的未知(即盲)线性组合。将Gibbs采样器和SI方法相结合，提出了一种新的计算效率较高的盲信号恢复算法。第二个项目涉及非参数分层贝叶斯分析。分别采用Gibbs采样器和SI进行敏感性分析和层次分析。要做到这一点，从理论上理解狄利克雷过程的数学性质是必要的。作为应用，还提出了贝叶斯非参数方法和SI过程，以解决估计人类繁殖力的问题，即可识别概念的每周期概率。第三个项目包含了一些关于如何更有效地使用多个推算数据集的想法。提出了一种“分裂抽样”的方法，对联合输入的完整数据进行混合，然后利用重要性权重进行调整。这将产生一个更有效的估计器，称为“交叉匹配”估计器，对感兴趣的数量进行估计。提出了比较序贯插值法和Gibbs采样法两种新的蒙特卡罗模拟方法进行贝叶斯统计分析的研究目标，并将这两种方法应用于几个重要问题。首先要解决的问题是数字通信中的一个问题，称为“盲解卷积”。它在地震学、水声学、多点网络等领域有着广泛的应用。我们计划使用一个完整的概率模型来描述系统，并应用前面提到的蒙特卡罗方法来克服计算困难。另一个应用是对人类繁殖力的估计，即可识别概念的每周期概率。由于夫妇之间等待怀孕的时间有相当大的差异，以前的工作主要集中在参数模型上来解释这种异质性并估计生育能力的群体分布。我们提出了另一种方法，它不要求分布遵循特定的参数形式。我们方法中的主要困难之一是计算。新的蒙特卡罗方法可以很好地应用。这种类型的“贝叶斯非参数问题”长期以来一直是理论统计学家的话题。近年来，蒙特卡罗方法在计算中的新发展使其应用性大大提高。我们的最后一个项目包含了一些关于如何更有效地使用蒙特卡罗样本的想法。人们早就认识到，我们的计算能力与我们的理论思维直接相关。新的计算方法给我们提出了许多有趣的理论问题。我们计划研究其中的一些问题，一个是关于两种方法的相对效率和数学性质，另一个是如何更好地利用从蒙特卡罗模拟获得的样本。

Liu 9404344 The method of sequential imputation (henceforth, SI) introduced in Kong, Liu and Wong (1994) is a special way of conducting multiple imputations in treating missing data problems and has shown potential in providing efficient computational methods for some difficult problems in genetic linkage analysis (e.g., Kong et. al. 1993; Irwin, Cox and Kong 1993). The method is generally applicable when the corresponding complete data (Bayesian) predictive distributions are easy and is especially useful when data are collected sequentially. The Gibbs sampler is a recently popular tool for sampling from the Bayesian posterior distributions to facilitate inference. The proposed research targets at comparing the relative merits of the two methods and combining the two methods in some applications. In the first project, the SI method is applied to attack the blind deconvolution problem in digital communication, where it is assumed that the observed signals are an unknown (thus blind) linear combination of discrete input signals. A new computationally efficient algorithm for blind signal restorations results from combining the Gibbs sampler and the SI method. The second project is concerned with nonparametric hierarchical Bayesian analysis. Both the Gibbs sampler and the SI are applied to conduct sensitivity analysis and hierarchical analysis. In doing these, some theoretical understanding of the mathematical properties of a Dirichlet process is necessary. As an application, the Bayesian nonparametric method and the SI procedure are also proposed to address the problem of estimating human fecundability, or the per-cycle probability of a recognizable conception. The third project contains some ideas on how to use multiply imputed data sets more efficiently. A "split sampling" method is proposed to hybrid the jointly imputed complete data and then to adjust by using importance weights. This will result in a more efficient estimator, called "cross-match" estimate, of the quantity of interest. T he proposed research targets at comparing two novel Monte Carlo simulation methods, the sequential imputation and the Gibbs sampler, for Bayesian statistical analysis, and applying the two methods to several important problems. The first problem to attack is one in digital comunication, called "blind deconvolution." It has many applications in, for example, seismology, underwater acoustics, and multipoint network. We plan to use a complete probabilistic model to describe the system and apply the aforementioned Monte Carlo methods to overcome computational difficulties. Another application is the estimation of human fecundability, or the per-cycle probability of a recognizable conception. As there is considerable variability among couples in their waiting times to pregnancy, previous work has focused on parametric models to account for this heterogeneity and to estimate the population distribution of fecundability. We propose an alternative approach that does not require that the distribution follow a particular parametric form. One of the major difficulty in our approach is computational. The new Monte Carlo methods can be suitably applied. This type of "Bayesian nonparametric problem" has long been a topic for theoretical statisticians. Recent development of novel Monte Carlo methods in computation boomed its applicability. Our final project contains some ideas on how to use Monte Carlo samples more efficiently. It has long been recognized that our computational ability is directly linked to our theoretical thinking. The new computational methods presents us many interesting theoretical questions. We plan to study some of these questions, one of which is concerned with the relative efficiencies and mathematical properties of the two methods, another is concerned with how we can make better use of the samples obtained from Monte Carlo simulations.