  Spike-and-slab and horseshoe regression are arguably the most popular
Bayesian variable selection approaches for linear regression models. However,
their performance can deteriorate if outliers and heteroskedasticity are
present in the data, which are common features in many real-world statistics
and machine learning applications. In this work, we propose a Bayesian
nonparametric approach to linear regression that performs variable selection
while accounting for outliers and heteroskedasticity. Our proposed model is an
instance of a Dirichlet process scale mixture model with the advantage that we
can derive the full conditional distributions of all parameters in closed form,
hence producing an efficient Gibbs sampler for posterior inference. Moreover,
we present how to extend the model to account for heavy-tailed response
variables. The performance of the model is tested against competing algorithms
on synthetic and real-world datasets.
