  In a variety of application areas, there is a growing interest in analyzing
high dimensional sparse count data, with sparsity exhibited by an
over-abundance of zeros and small non-zero counts. Existing approaches for
analyzing multivariate count data via Poisson or negative binomial log-linear
hierarchical models with zero-inflation cannot flexibly adapt to the level and
nature of sparsity in the data. We develop a new class of continuous
local-global shrinkage priors tailored for sparse counts. Theoretical
properties are assessed, including posterior concentration, stronger control on
false discoveries in multiple testing, robustness in posterior mean and
super-efficiency in estimating the sampling density. Simulation studies
illustrate excellent small sample properties relative to competitors. We apply
the method to detect rare mutational hotspots in exome sequencing data and to
identify cities most impacted by terrorism.
