Bayes Shrinkage at GWAS scale: Convergence and Approximation Theory of a Scalable MCMC Algorithm for the Horseshoe Prior

Understand

The horseshoe prior is frequently employed in Bayesian analysis of high-dimensional models, and has been shown to achieve minimax optimal risk properties when the truth is sparse.

• While optimization-based algorithms for the extremely popular Lasso and elastic net procedures can scale to dimension in the hundreds of thousands, algorithms for the horseshoe that use Markov chain Monte Carlo (MCMC) for computation are limited to problems an order of magnitude smaller.
• This is due to high computational cost per step and growth of the variance of time-averaging estimators as a function of dimension.
• We propose two new MCMC algorithms for computation in these models that have improved performance compared to existing alternatives.