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Regular version of the site

Article

Bayesian inference for spectral projectors of the covariance matrix

Electronic journal of statistics. 2018. Vol. 12. No. 1. P. 1948-1987.
Silin I., Spokoiny V.

Let X_1,…,X_n be an i.i.d. sample in R^p with zero mean and the covariance matrix \Sigma^{*}. The classical PCA approach recovers the projector \P_J^{*} onto the principal eigenspace of \Sigma^{*} by its empirical counterpart \hat \P_J. Recent paper [24] investigated the asymptotic distribution of the Frobenius distance between the projectors \|\hat \P_J - \P_J^{*}\|_2, while [27] offered a bootstrap procedure to measure uncertainty in recovering this subspace \P_J^{*} even in a finite sample setup. The present paper considers this problem from a Bayesian perspective and suggests to use the credible sets of the pseudo-posterior distribution on the space of covariance matrices induced by the conjugated Inverse Wishart prior as sharp confidence sets. This yields a numerically efficient procedure. Moreover, we theoretically justify this method and derive finite sample bounds on the corresponding coverage probability. Contrary to [24, 27], the obtained results are valid for non-Gaussian data: the main assumption that we impose is the concentration of the sample covariance \hat \Sigma in a vicinity of \Sigma^{*}. Numerical simulations illustrate good performance of the proposed procedure even on non-Gaussian data in a rather challenging regime.