Accelerating Bayesian Synthetic Likelihood With the Graphical Lasso

ABSTRACT

Simulation-based Bayesian inference methods are useful when the statistical model of interest does not possess a computationally tractable likelihood function. One such likelihood-free method is approximate Bayesian computation (ABC), which approximates the likelihood of a carefully chosen summary statistic via model simulation and nonparametric density estimation. ABC is known to suffer a curse of dimensionality with respect to the size of the summary statistic. When the model summary statistic is roughly normally distributed in regions of the parameter space of interest, Bayesian synthetic likelihood (BSL), which uses a normal likelihood approximation for a summary statistic, is a useful method that can be more computationally efficient than ABC. However, BSL requires estimation of the covariance matrix of the summary statistic for each proposed parameter, which requires a large number of simulations to estimate precisely using the sample covariance matrix when the summary statistic is high dimensional. In this article, we propose to use the graphical lasso to provide a sparse estimate of the precision matrix. This approach can estimate the covariance matrix accurately with significantly fewer model simulations. We discuss the nontrivial issue of tuning parameter choice in the context of BSL and demonstrate on several complex applications that our method, which we call BSLasso, provides significant improvements in computational efficiency whilst maintaining the ability to produce similar posterior distributions to BSL. The BSL and BSLasso methods applied to the examples of this article are implemented in the $\mathit{BSL}$ package in R, which is available on the Comprehensive R Archive Network. Supplemental materials for this article are available online.