TABLE 1.

Overview of properties of co‐data learnt methods compared in Section 5: group lasso obtains group sparsity by penalizing the regression coefficients with a group lasso penalty governed by one global penalty parameter; GRridge uses empirical Bayes moment estimation to obtain group‐adaptive ridge penalties; graper employs a full Bayes model with group‐specific spike‐and‐slab priors; gren uses an empirical‐variational Bayes approach for group‐adaptive elastic net penalties; ecpc combines empirical Bayes moment estimation for group‐adaptive ridge penalties with shrinkage on the group level to account for various co‐data

		Method
Property		group lasso 4 , 5 , 7	GRridge 8	graper 9	gren 10	ecpc
Group‐adaptive		‐	v	v	v	v
Type of covariate model:	Dense	‐	v	v	v	v
	Group‐sparse	v	‐	‐	‐	v
	Sparse	v/‐${}^a$	v/‐${}^b$	v	v	v/‐${}^b$
Type of co‐data:	Non‐overlapping groups	v	v	v	v	v
	Overlapping groups	v	v	‐	‐	v
	Hierarchical groups	v	‐	‐	‐	v
	Multiple co‐data sources	‐	v	‐	v	v
Hyperparameter shrinkage (many groups)		‐	‐	v/‐${}^c$	‐	v
Type of response model:	Linear	v	v	v	‐	v
	Binary	v	v	v	v	v
	Survival	v	v	‐	‐	v

${}^{\mathrm{a}}$Can be accommodated but may lead to inferior performance. ⁸ , ⁹

${}^{\mathrm{b}}$Using posterior selection.

${}^{\mathrm{c}}$Graper uses a vague hyperprior on the hyperparameters.