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Abstract:
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Zhang et al. (2020) introduce the R2D2 prior, which imposes a prior on the coefficient of determination $R^2$ and propagates uncertainty on the regression coefficients via a Dirichlet decomposition. However, this prior and other shrinkage priors so far have only been developed for single-level models such as linear regression. Our work generalizes and extends the R2D2 prior to the case of multilevel models, leading to what we call the R2D2M2 prior. The proposed prior comes with interpretable hyperparameters, which are intuitively related to desired a-priori levels of shrinkage. Moreover, our prior jointly spans the whole set of additive regression terms to enable global regularization of the model. We implement in Stan and also derive a collapsed block Gibbs sampler. In extensive experiments on simulated and real data, we show that our prior is well calibrated and has desirable global and local regularization properties. Finally, we offer guidelines on the choice of the priors' hyperparameters and discuss the practical implications of our developed methods.
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