Seemingly Shallow Priors Are Sufficiently Expressive for Asymptotically Bayes Optimal Multiple Testing under Sparsity

One-group global-local shrinkage priors, popular alternatives to two-groups spike-and-slab priors, are widely used in Bayesian inference. Regarding the asymptotic optimality of high-dimensional Bayesian multiple testing procedures, the dominant view is that one-group priors should have multiple hyperparameters and heavy tails so that procedures based on these priors are asymptotically Bayes optimal under sparsity (ABOS). Liang et al. (2026) note that virtually all previous procedures based on one-group priors rely solely on the posterior. Instead, by considering both the prior and posterior, Liang et al. (2026) show that relative belief procedures using a simple one-group light-tailed normal prior and a one-group uniform prior, each with a single hyperparameter, are sufficient to be ABOS. All previous ABOS properties of procedures based on one-group priors are established under additive symmetric 0-1 loss. In this paper, we generalize the ABOS properties of relative belief procedures under additive weighted 0-1 loss and broader asymptotic regimes. A rich and flexible scale family of one-group priors, each with a single hyperparameter, can be used in our generalized relative belief procedures, ranging from light-tailed to heavy-tailed priors. The priors can be symmetric or asymmetric, and they can have bounded or unbounded support. Arguably, the ABOS properties of our generalized relative belief procedures are more general and stronger than any existing ABOS properties of procedures based on global-local priors.