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

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  • Abrahamsen, Tavis
  • Hobert, James P.

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.

Suggested Citation

  • Abrahamsen, Tavis & Hobert, James P., 2019. "Fast Monte Carlo Markov chains for Bayesian shrinkage models with random effects," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 61-80.
  • Handle: RePEc:eee:jmvana:v:169:y:2019:i:c:p:61-80
    DOI: 10.1016/j.jmva.2018.08.014
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    References listed on IDEAS

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    1. Johnson, Alicia A. & Jones, Galin L., 2015. "Geometric ergodicity of random scan Gibbs samplers for hierarchical one-way random effects models," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 325-342.
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