Bayesian inference for spectral projectors of covariance matrix

Let X1; : : : ;Xn be i.i.d. sample in Rp with zero mean and the covariance matrix . The classic principal component analysis esti- mates the projector P J onto the direct sum of some eigenspaces of by its empirical counterpart bPJ . Recent papers [20, 23] investigate the asymptotic distribution of the Frobenius distance between the projectors k bPJ ??P J k2 . The problem arises when one tries to build a condence set for the true projector eectively. 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 con- centration of the sample covariance b in a vicinity of . The obtained results are applied to construction of sharp condence sets for the true pro- jector. Numerical simulations illustrate good performance of the proposed procedure even on non-Gaussian data in quite challenging regime.