A Laplace Mixture Representation of the Horseshoe and Some Implications

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