Abstract
Gibbs sampling is a widely used Markov chain Monte Carlo (MCMC) method for numerically approximating integrals of interest in Bayesian statistics and other mathematical sciences. Many implementations of MCMC methods do not extend easily to parallel computing environments, as their inherently sequential nature incurs a large synchronization cost. In the case study illustrated by this paper, we show how to do Gibbs sampling in a fully data-parallel manner on a graphics processing unit, for a large class of exchangeable models that admit latent variable representations. Our approach takes a systems perspective, with emphasis placed on efficient use of compute hardware. We demonstrate our method on a Horseshoe Probit regression model and find that our implementation scales effectively to thousands of predictors and millions of data points simultaneously.
Highlights
The Bayesian statistical paradigm has a variety of desirable properties
We present a case study of a way to implement Markov chain Monte Carlo (MCMC) for a large class of Bayesian models that admit exchangeable likelihoods with latent variable representations
Preliminary descriptive analysis narrowed the available independent variables down to a set of p = 141 interesting predictors—where interesting was determined according to signal-to-noise ratios in maximum-likelihood estimation— and we used our algorithm to fit a Horseshoe Probit regression model to the resulting data set, to see how many of the interesting predictors survived the regularization process imposed by the Horseshoe prior
Summary
The Bayesian statistical paradigm has a variety of desirable properties It accounts for the uncertainty inherent in statistical inference by producing a posterior distribution, which fundamentally contains more information about the unknown quantities of interest than a point estimate. In the sections that follow, we describe GPUs, characterize models in which this approach is usable, and demonstrate the method on a Horseshoe Probit model with N = 1,000,000 and p = 1000. Standard computation with such N may take O (days)—the method we describe runs in O (minutes). Statistics and Computing (2019) 29:301–310 menting GPU Gibbs sampling, in the context of a Horseshoe Probit regression model
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