  We study multiple testing in the normal means problem with estimated
variances that are shrunk through empirical Bayes methods. The situation is
asymmetric in that a prior is posited for the nuisance parameters (variances)
but not the primary parameters (means). If the prior were known, one could
proceed by computing p-values conditional on sample variances; a strategy
called partially Bayes inference by Sir David Cox. These conditional p-values
satisfy a Tweedie-type formula and are approximated at nearly-parametric rates
when the prior is estimated by nonparametric maximum likelihood. If the
variances are in fact fixed, the approach retains type-I error guarantees.
