Statistical modeling and inference for high-dimensional biological data

由于复杂的依赖结构和参数空间的高维性，传统的统计方法往往难以应用于复杂的生物数据。在本提案中，我们的目标是对如何开发专门针对高维相关数据的统计推断方法和模型选择方法进行理论研究。由于完整数据的确切似然往往难以计算，复合似然方法（Lindsay, 1988, Cox and Reid, 2004, Varin and Vidoni， 2005等）被提出作为一种替代方法，使用在数据的不同子集上定义的有效似然对象的组合来描述数据。在存在缺失观测值的情况下，我们建议开发一种新的COMP-EM算法，其中每个对数似然对象根据相应子集中的观测数据采取单独的条件期望。当样本量很小时，最大似然方法的一个有用的替代方法是为参数添加先验分布，并计算感兴趣参数的后验分布。我们提出了类似于通常的复合似然定义的复合后验分布的概念，并研究了复合贝叶斯框架下多重归算技术和改进贝叶斯信息准则的理论性质。关于潜在的应用，我们将研究如何使用上述提出的方法来分析基因网络构建的概率模型。为了进一步确定最佳网络拓扑，我们需要采用基于复合似然的有效模型选择标准。总之，这个高维生物数据建模项目不仅涉及对各种统计问题的方法调查，而且旨在为分析具有挑战性的生物问题提供新的统计工具。

In the presence of complicated dependency structure and large dimensionality of the parameter space, it is often prohibitively difficult to apply traditional statistical methods on complex biological data. In this proposal, we aim to conduct theoretical investigations on how to develop statistical inference methods and model selection approaches especially designed for high-dimensional correlated data. As the exact likelihood of the full data is often computationally intractable, composite likelihood method (Lindsay, 1988, Cox and Reid, 2004, Varin and Vidoni, 2005, etc) has been proposed as an alternative way to describe the data using a combination of valid likelihood objects defined on different subsets of the data. In the presence of missing observations, we propose to develop a new COMP-EM algorithm in which each log likelihood object takes separate conditional expectation based on the observed data in the corresponding subset. When sample sizes are small, a useful alternative approach to the maximum likelihood approach is to add a prior distribution for the parameters and compute the posterior distribution of the parameters of interest. We propose to introduce the concept of composite posterior distribution analogous to the definition of the usual composite likelihood and investigate the theoretical properties of the multiple imputation technique and the modified Bayesian Information Criterion under the composite Bayes framework. With regard to the potential application, we will investigate how to employ the proposed methodologies developed above to analyze probabilistic models for gene network construction. To further determine the best network topology, we need to employ valid model selection criteria based on the composite likelihood. In conclusion, this project of modelling high-dimensional biological data involves not only methodological investigations on various statistical problems but also aims at providing new statistical tools for analyzing challenging biological problems.