  In this paper we study the asymptotic properties of Bayesian multiple testing
procedures for a large class of Gaussian scale mixture pri- ors. We study two
types of multiple testing risks: a Bayesian risk proposed in Bogdan et al.
(2011) where the data are assume to come from a mixture of normal, and a
frequentist risk similar to the one proposed by Arias-Castro and Chen (2017).
Following the work of van der Pas et al. (2016), we give general conditions on
the prior such that both risks can be bounded. For the Bayesian risk, the bound
is almost sharp. This result show that under these conditions, the considered
class of continuous prior can be competitive with the usual two-group model
(e.g. spike and slab priors). We also show that if the non-zeros component of
the parameter are large enough, the minimax risk can be made asymptotically
null. The separation rates obtained are consistent with the one that could be
guessed from the existing literature (see van der Pas et al., 2017b). For both
problems, we then give conditions under which an adaptive version of the result
can be obtained.
