  In the sparse normal means model, convergence of the Bayesian posterior
distribution associated to spike and slab prior distributions is considered.
The key sparsity hyperparameter is calibrated via marginal maximum likelihood
empirical Bayes. The plug-in posterior squared-$L^2$ norm is shown to converge
at the minimax rate for the euclidean norm for appropriate choices of spike and
slab distributions. Possible choices include standard spike and slab with heavy
tailed slab, and the spike and slab LASSO of Rockov\'a and George with heavy
tailed slab. Surprisingly, the popular Laplace slab is shown to lead to a
suboptimal rate for the full empirical Bayes posterior. This provides a
striking example where convergence of aspects of the empirical Bayes posterior
does not entail convergence of the full empirical Bayes posterior itself.
