  The adoption of continuous shrinkage priors in high-dimensional linear models
has gained momentum, driven by their theoretical and practical advantages. One
of these shrinkage priors is the R2D2 prior, which comes with intuitive
hyperparameters and well understood theoretical properties. The core idea is to
specify a prior on the percentage of explained variance $R^2$ and to conduct a
Dirichlet decomposition to distribute the explained variance among all the
regression terms of the model. Due to the properties of the Dirichlet
distribution, the competition among variance components tends to gravitate
towards negative dependence structures, fully determined by the individual
components' means. Yet, in reality, specific coefficients or groups may compete
differently for the total variability than the Dirichlet would allow for. In
this work we address this limitation by proposing a generalization of the R2D2
prior, which we term the Generalized Decomposition R2 (GDR2) prior.
  Our new prior provides great flexibility in expressing dependency structures
as well as enhanced shrinkage properties. Specifically, we explore the
capabilities of variance decomposition via logistic normal distributions.
Through extensive simulations and real-world case studies, we demonstrate that
GDR2 priors yield strongly improved out-of-sample predictive performance and
parameter recovery compared to R2D2 priors with similar hyper-parameter
choices.
