  A reciprocal LASSO (rLASSO) regularization employs a decreasing penalty
function as opposed to conventional penalization approaches that use increasing
penalties on the coefficients, leading to stronger parsimony and superior model
selection relative to traditional shrinkage methods. Here we consider a fully
Bayesian formulation of the rLASSO problem, which is based on the observation
that the rLASSO estimate for linear regression parameters can be interpreted as
a Bayesian posterior mode estimate when the regression parameters are assigned
independent inverse Laplace priors. Bayesian inference from this posterior is
possible using an expanded hierarchy motivated by a scale mixture of double
Pareto or truncated normal distributions. On simulated and real datasets, we
show that the Bayesian formulation outperforms its classical cousin in
estimation, prediction, and variable selection across a wide range of scenarios
while offering the advantage of posterior inference. Finally, we discuss other
variants of this new approach and provide a unified framework for variable
selection using flexible reciprocal penalties. All methods described in this
paper are publicly available as an R package at:
https://github.com/himelmallick/BayesRecipe.
