IRTG1792DP2018 027
Bayesian inference for spectral projectors of covariance matrix
Igor Silin
Vladimir Spokoiny
Abstract
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.
Keywords:
covariance matrix, spectral projector, principal
component analysis, Bernstein { von Mises theorem.
JEL classification: