On the local shrinkage of half-cauchy priors

The half-Cauchy prior has gained widespread recognition and adoption in Bayesian statistics, especially recently where it is applied to the scale parameter in some hierarchical representations of scale-mixture of normals (SMN) models. It is continued relevance stems from its ability to both provide a robust regularization while at the same time allow flexible model estimation, even high-dimensional settings. An analysis of the family of half-Cauchy priors is provided. It is shown that this model achieves posterior consistency under some assumptions. The MCEM algorithm is calculated from a hierarchical representation of this family and its performance is compared to other priors using simulated and real data and is shown to perform well.