Statistical modeling and inference for high-dimensional biological data

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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.