  We consider the problem of subspace estimation in a Bayesian setting. Since
we are operating in the Grassmann manifold, the usual approach which consists
of minimizing the mean square error (MSE) between the true subspace $U$ and its
estimate $\hat{U}$ may not be adequate as the MSE is not the natural metric in
the Grassmann manifold. As an alternative, we propose to carry out subspace
estimation by minimizing the mean square distance (MSD) between $U$ and its
estimate, where the considered distance is a natural metric in the Grassmann
manifold, viz. the distance between the projection matrices. We show that the
resulting estimator is no longer the posterior mean of $U$ but entails
computing the principal eigenvectors of the posterior mean of $U U^{T}$.
Derivation of the MMSD estimator is carried out in a few illustrative examples
including a linear Gaussian model for the data and a Bingham or von Mises
Fisher prior distribution for $U$. In all scenarios, posterior distributions
are derived and the MMSD estimator is obtained either analytically or
implemented via a Markov chain Monte Carlo simulation method. The method is
shown to provide accurate estimates even when the number of samples is lower
than the dimension of $U$. An application to hyperspectral imagery is finally
investigated.
