A Laplace Mixture Representation of the Horseshoe and Some Implications

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.