A Laplace Mixture Representation of the Horseshoe and Some Implications

TL;DR

This paper introduces a new Laplace mixture representation of the horseshoe prior, revealing its properties and implications for Bayesian sparse signal recovery and penalized regression algorithms.

Contribution

It provides an explicit Laplace mixture representation of the horseshoe density, establishing its properties and linking it to optimization algorithms for sparse estimation.

Findings

Horseshoe density is a scale mixture of Laplace densities.

The representation proves the density's complete monotonicity.

The approach confirms the sparsity of the resulting estimates.

Abstract

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.