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.