Title: Bayesian inference for spectral projectors of covariance matrix,
Abstract: Let $X_1, \ldots, X_n$ be i.i.d. sample in $\mathbb{R}^p$ with zero mean and
the covariance matrix $\mathbf{\Sigma^*}$. The classic principal component
analysis estimates the projector $\mathbf{P^*_{\mathcal{J}}}$ onto the direct
sum of some eigenspaces of $\mathbf{\Sigma^*}$ by its empirical counterpart
$\mathbf{\widehat{P}_{\mathcal{J}}}$. Recent papers [Koltchinskii, Lounici
(2017)], [Naumov et al. (2017)] investigate the asymptotic distribution of the
Frobenius distance between the projectors $\|
\mathbf{\widehat{P}_{\mathcal{J}}} - \mathbf{P^*_{\mathcal{J}}} \|_2$. The
problem arises when one tries to build a confidence set for the true projector
effectively. We consider the problem from Bayesian perspective and derive an
approximation for the posterior distribution of the Frobenius distance between
projectors. The derived theorems hold true for non-Gaussian data: the only
assumption that we impose is the concentration of the sample covariance
$\mathbf{\widehat{\Sigma}}$ in a vicinity of $\mathbf{\Sigma^*}$. The obtained
results are applied to construction of sharp confidence sets for the true
projector. Numerical simulations illustrate good performance of the proposed
procedure even on non-Gaussian data in quite challenging regime.