Geometric Ergodicity of Gibbs Algorithms for a Normal Model With a Global-Local Shrinkage Prior

We consider Gibbs samplers for a normal linear regression model with a global-local shrinkage prior and show that they produce geometrically ergodic Markov chains. First, under the horseshoe local prior and a three-parameter beta global prior under some assumptions, we prove geometric ergodicity for a Gibbs algorithm in which it is relatively easy to update the global shrinkage parameter. Second, we consider a more general class of global-local shrinkage priors. Under milder conditions, geometric ergodicity is proved for two- and three-stage Gibbs samplers based on rejection sampling. We also construct a practical rejection sampling method in the horseshoe case. Finally, a simulation study is performed to compare proposed and existing methods.