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Bayesian inference for spectral projectors of covariance matrix. (arXiv:1711.11532v1 [math.ST])
来源于:arXiv
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
$\widehat{\mathbf{P}}_{\mathcal{J}}$. Recent papers [Koltchinskii, Lounici,
2017], [Naumov et al., 2017] investigate the asymptotic distribution of the
Frobenius distance between the projectors $\|
\widehat{\mathbf{P}}_{\mathcal{J}} - \mathbf{P}^*_{\mathcal{J}} \|_2$. The
problem arises when one tries to build a confidence set for the true projector
effectively. We consider the problem from a 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 sample covariance
$\widehat{\mathbf{\Sigma}}$ in a vicinity 查看全文>>