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 cel...

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Bibliographic Details
Published in:IEEE signal processing letters 2022, Vol.29, p.2547-2551
Main Authors: Sagar, Ksheera, Bhadra, Anindya
Format: Article
Language:English
Subjects:
Algorithms
Bayesian sparse signal recovery
Bernstein–Widder
completely monotone function
Concavity
Convergence
Density
Estimation
Linear approximation
local linear approximation
Mathematical models
Mixtures
Optimization
Random variables
Representations
Signal reconstruction
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Summary: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.
ISSN:1070-9908
1558-2361
DOI:10.1109/LSP.2022.3228491