The R2D2 Prior for Generalized Linear Mixed Models

TL;DR

This paper introduces a novel Bayesian prior based on the model fit measure $R^2$, which induces priors on parameters in generalized linear mixed models, enhancing flexibility and computational ease.

Contribution

It proposes a new prior on $R^2$ that induces priors on model parameters, with closed-form solutions and approximation strategies for practical implementation.

Findings

Performs well in high-dimensional settings

Effective in modeling random effects

Provides flexible prior construction

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 $R^2$ 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.