Geometric ergodicity for Bayesian shrinkage models

In recent years, a large variety of continuous shrinkage priors have been developed for a Bayesian analysis of the standard linear regression model in high dimensional settings. We consider two such priors, the Dirichlet-Laplace prior (developed in Bhattacharya et al. (2013)), and the Normal-Gamma prior (developed in (Griffin and Brown, 2010)). For both Dirichlet-Laplace and Normal-Gamma priors, Gibbs sampling Markov chains have been developed to generate approximate samples from the corresponding posterior distributions. We show by using a drift and minorization based analysis that the Gibbs sampling Markov chains corresponding to the aforementioned models are geometrically ergodic. Establishing geometric ergodicity of these Markov chains is crucial, as it provides theoretical justification for the use of Markov chain CLT, which can then be used to obtain asymptotic standard errors for Markov chain based estimates of posterior quantities. Both Gibbs samplers in the paper use the Generalized Inverse Gaussian (GIG) distribution, as one of the conditional distributions. A novel contribution of our convergence analysis is the use of drift functions which include terms that are negative fractional powers of normal random variables, to tackle the presence of the GIG distribution.