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 Φ 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 Φβ=b has the desired Gaussian distribution. We can then solve the linear system by the conjugate gradient (CG) algorithm through matrix-vector multiplications by Φ; this involves no explicit factorization or calculation of Φ 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 n=72,489 patients and p=22,175 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.