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

Kong， Liu和Wong（1994）提出的顺序归算方法（以下简称SI）是一种处理缺失数据问题进行多重归算的特殊方法，在为遗传连锁分析中的一些难题提供高效的计算方法方面显示出潜力（例如，Kong et. al. 1993; Irwin, Cox and Kong 1993）。该方法一般适用于相应的完整数据（贝叶斯）预测分布容易的情况，尤其适用于顺序收集数据的情况。吉布斯采样器是最近流行的一种工具，用于从贝叶斯后验分布中采样以促进推理。本文的研究目标是比较两种方法的相对优点，并将两种方法在某些应用中结合起来。在第一个项目中，SI方法被应用于攻击数字通信中的盲反卷积问题，其中假设观察到的信号是离散输入信号的未知（因此是盲的）线性组合。将吉布斯采样器与SI法相结合，提出了一种计算效率高的盲信号恢复算法。第二个项目是关于非参数层次贝叶斯分析。采用Gibbs采样器和SI进行灵敏度分析和层次分析。在做这些时，对狄利克雷过程的数学性质有一些理论上的了解是必要的。作为一种应用，贝叶斯非参数方法和SI过程也被提出来解决估计人类可生育能力的问题，或一个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输入数据集的想法。提出了一种“分裂抽样”方法，将联合输入的完整数据进行混合，然后利用重要权值进行调整。这将导致对感兴趣的数量的更有效的估计，称为“交叉匹配”估计。本文提出的研究目标是比较贝叶斯统计分析的两种新的蒙特卡罗模拟方法——序列imputation和Gibbs sampler，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“盲反卷积”。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生育能力的估计，或一个可识别概念的每周期概率。由于夫妇在等待怀孕的时间上存在相当大的差异，以前的工作主要集中在参数模型上，以解释这种异质性并估计生育能力的人口分布。我们提出了一种替代方法，它不要求分布遵循特定的参数形式。我们的方法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。这类“贝叶斯非参数问题”一直是理论统计学家的研究课题。新的蒙特卡罗方法在计算方面的新发展使其适用性大大增强。我们的最终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，其中一个是关于两种方法的相对效率和数学性质，另一个是关于我们如何更好地利用从蒙特卡罗模拟中获得的样本。

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.