  Consider the Gaussian sequence model under the additional assumption that a
fixed fraction of the means is known. We study the problem of variance
estimation from a frequentist Bayesian perspective. The maximum likelihood
estimator (MLE) for $\sigma^2$ is biased and inconsistent. This raises the
question whether the posterior is able to correct the MLE in this case. By
developing a new proving strategy that uses refined properties of the posterior
distribution, we find that the marginal posterior is inconsistent for any
i.i.d. prior on the mean parameters. In particular, no assumption on the decay
of the prior needs to be imposed. Surprisingly, we also find that consistency
can be retained for a hierarchical prior based on Gaussian mixtures. In this
case we also establish a limiting shape result and determine the limit
distribution. In contrast to the classical Bernstein-von Mises theorem, the
limit is non-Gaussian. We show that the Bayesian analysis leads to new
statistical estimators outperforming the correctly calibrated MLE in a
numerical simulation study.
