  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.
