The estimation of quadratic functions of a multivariate normal mean is an inferential problem which, while being simple to state and often encountered in practice, leads to surprising complications both from frequentist and Bayesian points of view. The drawbacks of Bayesian inference using the constant noninformative prior are now well established and we consider in this paper the advantages and the shortcomings of alternative noninformative priors. We take into account frequentist coverage probability of confidence sets arising from these priors. Lastly, we derive some optimality properties of the associated Bayes estimators in the special case of independent components under quadratic loss. Postscript File (686kB)