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 the talk, we focus on the Bayesian lasso model developed by Park and Casella (2008) and generalized version – the Normal-Gamma model developed by Griffin and Brown (2010).

For both these models, a three-block Gibbs sampling algorithm to sample from the resulting intractable posterior distribution has been developed in Park and Casella and Griffin and Brown. We develop an alternative two-block Gibbs sampling algorithm for each model, 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.