Geometric Ergodicity of a Gibbs Algorithm for a Normal Model With a Horseshoe Prior

In this paper, we consider a two-stage Gibbs sampler for a normal linear regression model with a horseshoe prior. Under some assumptions, we show that it produces a geometrically ergodic Markov chain. In particular, we prove geometric ergodicity under some 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.