Bayesian Variable Selection in Structured HighDimensional Covariate Spaces with Applications in Genomics

We consider the problem of variable selection in regression modeling in high dimensional spaces where there is known structure among the covariates. This is an unconventional vari-able selection problem for two reasons: (1) The dimension of the covariate space is compa-rable, and often much larger, than the number of subjects in the study, and (2) the covariate space is highly structured, and in some cases it is desirable to incorporate this structural in-formation in to the model building process. We approach this problem through the Bayesian variable selection framework, where we assume that the covariates lie on an undirected graph and formulate an Ising prior on the model space for incorporating structural information. Cer-tain computational and statistical problems arise that are unique to such high dimensional, structured settings, the most interesting being the phenomenon of phase transitions. We pro-pose theoretical and computational schemes to mitigate these problems. We illustrate our methods on two different graph structures: the linear chain and the regular graph of degree 푘. Finally, we use our methods to study a specific application in genomics: the modeling of transcription factor binding sites in DNA sequences