We propose a new class of priors for linear regression, the R-square inducedDirichlet Decomposition (R2-D2) prior. The prior is induced by a Beta prior onthe coefficient of determination, and then the total prior variance of theregression coefficients is decomposed through a Dirichlet prior. We demonstrateboth theoretically and empirically the advantages of the R2-D2 prior over anumber of common shrink- age priors, including the Horseshoe, Horseshoe+, andDirichlet-Laplace priors. The R2-D2 prior possesses the fastest concentrationrate around zero and heaviest tails among these common shrinkage priors, whichis established based on its marginal density, a Meijer G-function. We show thatits Bayes estimator converges to the truth at a Kullback-Leiblersuper-efficient rate, attaining a sharper information theoretic bound thanexisting common shrinkage priors. We also demonstrate that the R2-D2 prioryields a consistent posterior. The R2-D2 prior permits straightforward Gibbssampling and thus enjoys computational tractability. The proposed prior isfurther investigated in a mouse gene expression application.
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