Abstract
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 Ročková and George with heavy tailed slab. Surprisingly, the popular Laplace slab is shown to lead to a suboptimal rate for the empirical Bayes posterior itself. This provides a striking example where convergence of aspects of the empirical Bayes posterior such as the posterior mean or median does not entail convergence of the complete empirical Bayes posterior itself.
Highlights
In the sparse normal means model, one observes a sequence X = (X1, . . . , Xn)Xi = θi + εi, i = 1, . . . , n, (1)with θ = (θ1, . . . , θn) ∈ Rn and ε1, . . . , εn i.i.d
We have developped a theory of empirical Bayes choice of the hyperparameter of spike and slab prior distributions
It extends the work of Johnstone and Silverman [12] in that here the complete empirical Bayes (EB) posterior distribution is considered
Summary
In a different spirit but still without using Dirac masses at 0, the paper [11] shows that, remarkably, it is possible to adopt an empirical Bayes approach on the entire unknown distribution function F of the vector θ, interpreting θ as sampled from a certain distribution, and the authors derive oracle results over p, p > 0, balls for the plug-in posterior mean (not including the case p = 0 though). We note the interesting work [20] that investigates necessary and sufficient conditions for sparse continuous priors to be rate-optimal The latter is for a fixed regularity parameter sn, while the results decribed in Section 2 (in particularity the suboptimality phenomenon, and upper-bounds using the empirical Bayes approach) are related to adaptation. 3. In Section 2.4, the spike and slab LASSO prior is considered and we provide a near-optimal adaptive rate for the corresponding complete empirical Bayes posterior distribution.
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