  This paper investigates a general family of covariance models with repeated
eigenvalues extending probabilistic principal component analysis (PPCA). A
geometric interpretation shows that these models are parameterised by flag
manifolds and stratify the space of covariance matrices according to the
sequence of eigenvalue multiplicities. The subsequent analysis sheds light on
PPCA and answers an important question on the practical identifiability of
individual eigenvectors. It notably shows that one rarely has enough samples to
fit a covariance model with distinct eigenvalues and that block-averaging the
adjacent sample eigenvalues with small gaps achieves a better
complexity/goodness-of-fit tradeoff.

Methodology