  The horseshoe prior, defined as a half Cauchy scale mixture of normal,
provides a state of the art approach to Bayesian sparse signal recovery. We
provide a new representation of the horseshoe density as a scale mixture of the
Laplace density, explicitly identifying the mixing measure. Using the
celebrated Bernstein--Widder theorem and a result due to Bochner, our
representation immediately establishes the complete monotonicity of the
horseshoe density and strong concavity of the corresponding penalty.
Consequently, the equivalence between local linear approximation and
expectation--maximization algorithms for finding the posterior mode under the
horseshoe penalized regression is established. Further, the resultant estimate
is shown to be sparse.
