  High-dimensional data, where the number of variables exceeds or is comparable
to the sample size, is now pervasive in many scientific applications. In recent
years, Bayesian shrinkage models have been developed as effective and
computationally feasible tools to analyze such data, especially in the context
of linear regression. In this paper, we focus on the Normal-Gamma shrinkage
model developed by Griffin and Brown. This model subsumes the popular Bayesian
lasso model, and a three-block Gibbs sampling algorithm to sample from the
resulting intractable posterior distribution has been developed by Griffin and
Brown. We consider an alternative two-block Gibbs sampling algorithm and
rigorously demonstrate its advantage over the three-block sampler by comparing
specific spectral properties. In particular, we show that the Markov operator
corresponding to the two-block sampler is trace class (and hence
Hilbert-Schmidt), whereas the operator corresponding to the three-block sampler
is not even Hilbert-Schmidt. The trace class property for the two-block sampler
implies geometric convergence for the associated Markov chain, which justifies
the use of Markov chain CLT's to obtain practical error bounds for MCMC based
estimates. Additionally, it facilitates theoretical comparisons of the
two-block sampler with sandwich algorithms which aim to improve performance by
inserting inexpensive extra steps in between the two conditional draws of the
two-block sampler.
