Project Summary Alzheimer?s disease (AD) is the most common neurodegenerative disease without a cure, and most cases are often diagnosed in the late stage of the disease. To advance our understanding of the initiation, progression, and etiology of such a disease, many genetic and genomic studies, including genome-wide association studies (GWAS), have been conducted, successfully identifying some associated single-nucleotide polymorphisms (SNPs). However, since many of these associated SNPs are either close to multiple genes or far away from any genes, identifying the responsible genes and thus understanding biological mechanisms with the disease remain most challenging. Of paramount importance is unraveling and understanding the regulatory roles of susceptible SNPs and genes for complex human diseases such as AD, so that treatment and prevention strategies can be developed. Given the urgent need of understanding the biological mechanisms of the disease, the PIs propose to develop powerful statistical and computational tools for accurate inference of gene regulatory networks for AD patients and healthy subjects. The proposed project consists of two interconnected components: reconstruction and inference of regulatory networks of the genes and SNPs. It will be centered on structures of regulatory networks, with particular effort focused on the accuracy of discovery and unbiased inference in high-dimensional situations, where model pa- rameters describing regulatory networks may greatly exceed the sample size. With regard to reconstruction of regulatory networks, the project will develop the constrained maximum likelihood method to identify directionality as well as strengths of pairwise causal relations, modeled by directed acyclic graphs (DAGs) among the genes and SNPs. With regard to network inference, the project will develop novel tests as formal inference methods for DAGs. On this ground, high-dimensional inference tools will be developed for detecting structural changes of multi- ple networks using the constrained likelihood tests. Moreover, linear and nonlinear causal relations will be studied. Computationally, innovative strategies will be developed based on the state-of-the-art optimization techniques, and be used for scalable analysis of large-scale networks.