Bayesian Learning with Structured Sparsity

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

A common premise held in high-dimensional data analysis is that only a small fraction of the covariates/variables is relevant. This is termed as the sparsity assumption meaning that the unknown parameter, which needs to be inferred from the data, is sparse. In many real world applications, there is in addition some side information on the structure dependence among elements of this high-dimensional parameter. For example, they may be grouped, ordered, or linked on a graph. Consequently such structure constraints should be incorporated into the ordinary sparsity assumption, which is what we call the structured sparsity. The scientific foci of this proposal are to develop novel Bayesian theory, methodology and computational tools to adaptively utilize structure information into statistical inference, such as variable selection, estimation, and dimension reduction. Many problems in modern statistics can be formulated as recovering some unknown high-dimensional parameter from the noisy data. In some real world applications, besides the observed data, there is a kind of side information termed as "structured sparsity" on the unknown parameter. The ultimate goal of this project is to develop novel theory, methodology and algorithms to adaptively utilize such side information for statistical inference. The statistical and computational challenges addressed in this project are a response to practical problems arising from the Principle Investigator's collaborative research on neuroimaging analysis, binoinformatics, and marketing science. The proposed methodological development and statistics research will result in useful methods for broader applications.