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

Abstract: In this paper, we consider Gibbs samplers for a normal linear regression model with a global-local shrinkage prior. We show that they produce geometrically ergodic Markov chains under some assumptions. In the first half of the paper, we prove geometric ergodicity under the horseshoe local prior and a three-parameter beta global prior which does not have a finite $(p / 5)$-th negative moment, where $p$ is the number of regression coefficients. This is in contrast to the case of a known general result which is applicable if the global parameter has a finite approximately $(p / 2)$-th negative moment. In the second half of the paper, we consider a more general class of global-local shrinkage priors. Geometric ergodicity is proved for two-stage and three-stage Gibbs samplers based on rejection sampling without assuming the negative moment condition.