  In this article, our goal is to develop a method for Bayesian model averaging
in linear regression models to accommodate heavier tailed error distributions
than the normal distribution. Motivated by the use of the Huber loss function
in presence of outliers, Park and Casella (2008) proposed the concept of the
Bayesian Huberized lasso, which has been recently developed and implemented by
Kawakami and Hashimoto (2023), with hyperbolic errors. Because the Huberized
lasso cannot enforce regression coefficients to be exactly zero, we propose a
fully Bayesian variable selection approach with spike and slab priors, that can
address sparsity more effectively. Furthermore, while the hyperbolic
distribution has heavier tails than a normal distribution, its tails are less
heavy in comparison to a Cauchy distribution.Thus, we propose a regression
model, with an error distribution that encompasses both hyperbolic and
Student-t distributions. Our model aims to capture the benefit of using Huber
loss, but it can also adapt to heavier tails, and unknown levels of sparsity,
as necessitated by the data. We develop an efficient Gibbs sampler with
Metropolis Hastings steps for posterior computation. Through simulation
studies, and analyses of the benchmark Boston housing dataset and NBA player
salaries in the 2022-2023 season, we show that our method is competitive with
various state-of-the-art methods.
