Fast Monte Carlo Markov chains for Bayesian shrinkage models with random effects

Tavis Abrahamsen and James P. Hobert

Journal of Multivariate Analysis, 2019, vol. 169, issue C, 61-80

Abstract: When performing Bayesian data analysis using a general linear mixed model, the resulting posterior density is almost always analytically intractable. However, if proper conditionally conjugate priors are used, there is a simple two-block Gibbs sampler that is geometrically ergodic in nearly all practical settings, including situations where p>n (Abrahamsen and Hobert, 2017). Unfortunately, the (conditionally conjugate) multivariate Gaussian prior on β does not perform well in the high-dimensional setting where p≫n. In this paper, we consider an alternative model in which the multivariate Gaussian prior is replaced by the normal-gamma shrinkage prior developed by Griffin and Brown (2010). This change leads to a much more complex posterior density, and we develop a simple MCMC algorithm for exploring it. This algorithm, which has both deterministic and random scan components, is easier to analyze than the more obvious three-step Gibbs sampler. Indeed, we prove that the new algorithm is geometrically ergodic in most practical settings.

Keywords: Bayesian shrinkage prior; Geometric drift condition; Geometric ergodicity; High-dimensional inference; Large p/small n; Markov chain Monte Carlo (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1016/j.jmva.2018.08.014

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