  This paper concerns the robust regression model when the number of predictors
and the number of observations grow in a similar rate. Theory for M-estimators
in this regime has been recently developed by several authors [El Karoui et
al., 2013, Bean et al., 2013, Donoho and Montanari, 2013].
  Motivated by the inability of M-estimators to successfully estimate the
Euclidean norm of the coefficient vector, we consider a Bayesian framework for
this model. We suggest a two-component mixture of normals prior for the
coefficients and develop a Gibbs sampler procedure for sampling from relevant
posterior distributions, while utilizing a scale mixture of normal
representation for the error distribution . Unlike M-estimators, the proposed
Bayes estimator is consistent in the Euclidean norm sense. Simulation results
demonstrate the superiority of the Bayes estimator over traditional estimation
methods.
