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Joint mean–covariance estimation via the horseshoe

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  • Li, Yunfan
  • Datta, Jyotishka
  • Craig, Bruce A.
  • Bhadra, Anindya

Abstract

Seemingly unrelated regression is a natural framework for regressing multiple correlated responses on multiple predictors. The model is very flexible, with multiple linear regression and covariance selection models being special cases. However, its practical deployment in genomic data analysis under a Bayesian framework is limited due to both statistical and computational challenges. The statistical challenge is that one needs to infer both the mean vector and the inverse covariance matrix, a problem inherently more complex than separately estimating each. The computational challenge is due to the dimensionality of the parameter space that routinely exceeds the sample size. We propose the use of horseshoe priors on both the mean vector and the inverse covariance matrix. This prior has demonstrated excellent performance when estimating a mean vector or inverse covariance matrix separately. The current work shows these advantages are also present when addressing both simultaneously. A full Bayesian treatment is proposed, with a sampling algorithm that is linear in the number of predictors. MATLAB code implementing the algorithm is freely available from github at https://github.com/liyf1988/HS_GHS. Extensive performance comparisons are provided with both frequentist and Bayesian alternatives, and both estimation and prediction performances are verified on a genomic data set.

Suggested Citation

  • Li, Yunfan & Datta, Jyotishka & Craig, Bruce A. & Bhadra, Anindya, 2021. "Joint mean–covariance estimation via the horseshoe," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    • Handle: RePEc:eee:jmvana:v:183:y:2021:i:c:s0047259x20302979
    DOI: 10.1016/j.jmva.2020.104716
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    References listed on IDEAS

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    1. Carlos M. Carvalho & Nicholas G. Polson & James G. Scott, 2010. "The horseshoe estimator for sparse signals," Biometrika, Biometrika Trust, vol. 97(2), pages 465-480.
    2. Anindya Bhadra & Bani K. Mallick, 2013. "Joint High-Dimensional Bayesian Variable and Covariance Selection with an Application to eQTL Analysis," Biometrics, The International Biometric Society, vol. 69(2), pages 447-457, June.
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    Cited by:

    1. Adam N. Smith & Jim E. Griffin, 2023. "Shrinkage priors for high-dimensional demand estimation," Quantitative Marketing and Economics (QME), Springer, vol. 21(1), pages 95-146, March.

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