The R2D2 Prior for Generalized Linear Mixed Models

Abstract

In Bayesian analysis, the selection of a prior distribution is typically done by considering each parameter in the model. While this can be convenient, in many scenarios it may be desirable to place a prior on a summary measure of the model instead. In this work, we propose a prior on the model fit, as measured by a Bayesian coefficient of determination ($R^2)$, which then induces a prior on the individual parameters. We achieve this by placing a beta prior on R² and then deriving the induced prior on the global variance parameter for generalized linear mixed models. We derive closed-form expressions in many scenarios and present several approximation strategies when an analytic form is not possible and/or to allow for easier computation. In these situations, we suggest approximating the prior by using a generalized beta prime distribution and provide a simple default prior construction scheme. This approach is quite flexible and can be easily implemented in standard Bayesian software. Lastly, we demonstrate the performance of the method on simulated and real-world data, where the method particularly shines in high-dimensional settings, as well as modeling random effects.

KEYWORDS:

• Bayesian modeling
• Coefficient of determination
• Generalized beta prime distribution
• Goodness-of-fit

Acknowledgments

The authors thank Brandon Feng for help with the derivation of the Weibull model.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

The authors thank the National Institutes of Health (R01ES031651-01) and King Abdullah University of Science and Technology (3800.2) for financial support.