  Substantial research on structured sparsity has contributed to analysis of
many different applications. However, there have been few Bayesian procedures
among this work. Here, we develop a Bayesian model for structured sparsity that
uses a Gaussian process (GP) to share parameters of the sparsity-inducing prior
in proportion to feature similarity as defined by an arbitrary positive
definite kernel. For linear regression, this sparsity-inducing prior on
regression coefficients is a relaxation of the canonical spike-and-slab prior
that flattens the mixture model into a scale mixture of normals. This prior
retains the explicit posterior probability on inclusion parameters---now with
GP probit prior distributions---but enables tractable computation via
elliptical slice sampling for the latent Gaussian field. We motivate
development of this prior using the genomic application of association mapping,
or identifying genetic variants associated with a continuous trait. Our
Bayesian structured sparsity model produced sparse results with substantially
improved sensitivity and precision relative to comparable methods. Through
simulations, we show that three properties are key to this improvement: i)
modeling structure in the covariates, ii) significance testing using the
posterior probabilities of inclusion, and iii) model averaging. We present
results from applying this model to a large genomic dataset to demonstrate
computational tractability.
