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Bayesian Regression Using a Prior on the Model Fit: The R2-D2 Shrinkage Prior

Author

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  • Yan Dora Zhang
  • Brian P. Naughton
  • Howard D. Bondell
  • Brian J. Reich

Abstract

Prior distributions for high-dimensional linear regression require specifying a joint distribution for the unobserved regression coefficients, which is inherently difficult. We instead propose a new class of shrinkage priors for linear regression via specifying a prior first on the model fit, in particular, the coefficient of determination, and then distributing through to the coefficients in a novel way. The proposed method compares favorably to previous approaches in terms of both concentration around the origin and tail behavior, which leads to improved performance both in posterior contraction and in empirical performance. The limiting behavior of the proposed prior is 1/x , both around the origin and in the tails. This behavior is optimal in the sense that it simultaneously lies on the boundary of being an improper prior both in the tails and around the origin. None of the existing shrinkage priors obtain this behavior in both regions simultaneously. We also demonstrate that our proposed prior leads to the same near-minimax posterior contraction rate as the spike-and-slab prior. Supplementary materials for this article are available online.

Suggested Citation

  • Yan Dora Zhang & Brian P. Naughton & Howard D. Bondell & Brian J. Reich, 2022. "Bayesian Regression Using a Prior on the Model Fit: The R2-D2 Shrinkage Prior," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 117(538), pages 862-874, April.
  • Handle: RePEc:taf:jnlasa:v:117:y:2022:i:538:p:862-874
    DOI: 10.1080/01621459.2020.1825449
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    Cited by:

    1. Eric Yanchenko & Howard D. Bondell & Brian J. Reich, 2024. "Spatial regression modeling via the R2D2 framework," Environmetrics, John Wiley & Sons, Ltd., vol. 35(2), March.
    2. David Kohns & Noa Kallioinen & Yann McLatchie & Aki Vehtari, 2024. "The ARR2 prior: flexible predictive prior definition for Bayesian auto-regressions," Papers 2405.19920, arXiv.org, revised May 2024.

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