Discussion to ’ Uncertainty Quantification for the Horseshoe ’ by Stéphanie van der Pas , Botond Szabó , and Aad van der Vaart ∗

The authors present a detailed analysis of the asymptotic frequentist properties of credible sets derived from posteriors with normal-linear measurement models and horseshoe priors. Although we disagree with the claim that “In Bayesian practice credible balls are nevertheless used as if they were confidence sets”, the results in the paper are important for identifying where the horseshoe priors are fragile asymptotically, and hence particularly dangerous in the non-asymptotic regimes more typical of the applied problems where sparse models are needed. One clarification we believe is warranted is that the horseshoe family of prior distributions does not encode sparsity as is typically interpreted. Instead of partitioning parameters into those that are zero and non-zero, the horseshoe priors actually separate parameters into those that are resolvable by measurements and those that are not. In particular, as with any model the horseshoe priors cannot be interpreted outside of the context of a particular likelihood (Gelman et al., 2017). Consequently the statement that “τ can be interpreted as the proportion of nonzero parameters, up to a logarithmic factor” is not quite true. Piironen and Vehtari (2017b; 2017c) demonstrate that the effects of τ in horseshoe priors are intimately related to the measurement variability σ, even for the simple normal-linear measurement model. Figure 6 of Piironen and Vehtari (2017c), for example, clearly illustrates that rescaling the data changes the impact of the horseshoe prior unless τ is scaled by σ, even with an oracle prior information about the true number of significant parameters, p0 = pn. In particular, the resolution threshold √ 2 log(n/pn) arising in the paper implicitly assumes that the measurement variability σ is equal to 1,