Prior-Preconditioned Conjugate Gradient Method for Accelerated Gibbs Sampling in “Large n, Large p” Bayesian Sparse Regression

In a modern observational study based on healthcare databases, the number of observations and of predictors typically range in the order of 10⁵–10⁶ and of 10⁴–10⁵. Despite the large sample size, data rarely provide sufficient information to reliably estimate such a large number of parameters. Sparse regression techniques provide potential solutions, one notable approach being the Bayesian method based on shrinkage priors. In the “large n and large p” setting, however, the required posterior computation encounters a bottleneck at repeated sampling from a high-dimensional Gaussian distribution, whose precision matrix (Formula presented.) is expensive to compute and factorize. In this article, we present a novel algorithm to speed up this bottleneck based on the following observation: We can cheaply generate a random vector b such that the solution to the linear system (Formula presented.) has the desired Gaussian distribution. We can then solve the linear system by the conjugate gradient (CG) algorithm through matrix-vector multiplications by (Formula presented.); this involves no explicit factorization or calculation of (Formula presented.) itself. Rapid convergence of CG in this context is guaranteed by the theory of prior-preconditioning we develop. We apply our algorithm to a clinically relevant large-scale observational study with (Formula presented.) patients and (Formula presented.) clinical covariates, designed to assess the relative risk of adverse events from two alternative blood anti-coagulants. Our algorithm demonstrates an order of magnitude speed-up in posterior inference, in our case cutting the computation time from two weeks to less than a day. Supplementary materials for this article are available online.