Bayesian inference for spectral projectors of the covariance matrix

Abstract

Let X1,…,Xn be an i.i.d. sample in Rp with zero mean and the covariance matrix Σ∗. The classical PCA approach recovers the projector P∗J onto the principal eigenspace of Σ∗ by its empirical counterpart ˆPJ. Recent paper [24] investigated the asymptotic distribution of the Frobenius distance between the projectors ∥ˆPJ−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 ˆΣ in a vicinity of Σ∗. Numerical simulations illustrate good performance of the proposed procedure even on non-Gaussian data in a rather challenging regime.