Title: Fast Monte Carlo Markov chains for Bayesian shrinkage models with random effects,
Abstract: When performing Bayesian data analysis using a general linear mixed model,
the resulting posterior density is almost always analytically intractable.
However, if proper conditionally conjugate priors are used, there is a simple
two-block Gibbs sampler that is geometrically ergodic in nearly all practical
settings, including situations where $p > n$ (Abrahamsen and Hobert, 2017).
Unfortunately, the (conditionally conjugate) multivariate normal prior on
$\beta$ does not perform well in the high-dimensional setting where $p \gg n$.
In this paper, we consider an alternative model in which the multivariate
normal prior is replaced by the normal-gamma shrinkage prior developed by
Griffin and Brown (2010). This change leads to a much more complex posterior
density, and we develop a simple MCMC algorithm for exploring it. This
algorithm, which has both deterministic and random scan components, is easier
to analyze than the more obvious three-step Gibbs sampler. Indeed, we prove
that the new algorithm is geometrically ergodic in most practical settings.