  We consider the estimation of an n-dimensional vector s from the noisy
element-wise measurements of $\mathbf{s}\mathbf{s}^T$, a generic problem that
arises in statistics and machine learning. We study a mismatched Bayesian
inference setting, where some of the parameters are not known to the
statistician. We derive the full exact analytic expression of the asymptotic
mean squared error (MSE) in the large system size limit for the particular case
of Gaussian priors and additive noise. From our formulas, we see that
estimation is still possible in the mismatched case; and also that the minimum
MSE (MMSE) can be achieved if the statistician chooses suitable parameters. Our
technique relies on the asymptotics of the spherical integrals and can be
applied as long as the statistician chooses a rotationally invariant prior.
