An Evidential Route to Asymptotic Bayes Optimality under Sparsity

From a statistical evidence perspective, we establish some asymptotic optimality properties of certain multiple testing rules based on the relative belief ratio (Evans, 2015). Under the two-groups model with an additive 0-1 loss and within a Bayesian decision theoretic asymptotic framework of Bogdan et al. (2011), we show that relative belief multiple testing rules induced by a simple one-group light-tailed normal prior with a single hyperparameter achieve the same asymptotic Bayes risk as the Bayes oracle benchmark. This risk is the minimum achievable in this asymptotic framework. Despite originating from a different starting point, the evidential relative belief approach enjoys oracle properties. The relative belief multiple testing approach is fundamentally different from existing Bayesian multiple testing procedures, virtually all induced by more complex heavy-tailed one-group global-local shrinkage priors using purely posterior-based inferences (Datta & Ghosh, 2013; Ghosh et al., 2016; Bhadra et al., 2017; Ghosh & Chakrabarti, 2017; Qin & Ghosh, 2025). By measuring statistical evidence via both the prior and posterior, the relative belief approach reveals an alternative new inferential paradigm for attaining asymptotic Bayes optimality under sparsity, one that does not rely on developing increasingly elaborate priors.

This work studies high-dimensional Bayesian multiple testing under the asymptotic decision theoretic framework of Bogdan et al. (2011). In the existing literature, spike-and-slab priors are typically regarded as the theoretical “gold standard” for modeling sparsity, while one-group global–local shrinkage priors (e.g., horseshoe-type priors) are favored computationally. There is a series of theoretical works to understand how well those procedures based on one-group global–local shrinkage priors can approximate the optimal two-groups performance in large-scale high-dimensional Bayesian multiple testing problems. Specifically, such works established that such heavy-tailed global–local priors can achieve nearly or exact Asymptotic Bayes Optimality under Sparsity (ABOS) in terms of Bayes risk. However, these priors generally involve multiple hyperparameters, heavy tails, and analytically intractable marginal densities. We challenge the prevailing belief that heavy tails and more than one hyperparameter are essential for attaining ABOS. We show that a single-parameter, light-tailed Gaussian one-group prior can achieve exact ABOS in both univariate and multivariate normal means problems under the same asymptotic framework. To the best of our knowledge, this is the first result establishing exact ABOS with a light-tailed prior of such simplicity. With the blessing of a better rate of convergence property, our relative belief procedure via the normal prior uniformly outperforms the horseshoe procedure in a comprehensive simulation study. This opens a new avenue to achieve ABOS using the one-group light-tailed prior. The key distinction lies in the inferential paradigm. Rather than relying on purely posterior procedures, we adopt the relative belief inference, measuring statistical evidence carefully via the ratio of posterior to prior density. By using the relative belief inference, exact ABOS can be established using arguably the most direct and simplest arguments, without the intricate posterior concentration machinery commonly used in the global–local priors literature.