Bayesian inference for spectral projectors of the covariance matrix

Abstract

Let $X_{1},\ldots ,X_{n}$ be an i.i.d. sample in $\mathbb{R}^{p}$ with zero mean and the covariance matrix ${\boldsymbol{\varSigma }^{*}}$. The classical PCA approach recovers the projector $\boldsymbol{P}^{*}_{\mathcal{J}}$ onto the principal eigenspace of ${\boldsymbol{\varSigma }^{*}}$ by its empirical counterpart $\widehat{\boldsymbol{P}}_{\mathcal{J}}$. Recent paper [24] investigated the asymptotic distribution of the Frobenius distance between the projectors $\|\widehat{\boldsymbol{P}}_{\mathcal{J}}-\boldsymbol{P}^{*}_{\mathcal{J}}\|_{2}$, while [27] offered a bootstrap procedure to measure uncertainty in recovering this subspace $\boldsymbol{P}^{*}_{\mathcal{J}}$ 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 $\widehat{\boldsymbol{\varSigma }}$ in a vicinity of ${\boldsymbol{\varSigma }^{*}}$. Numerical simulations illustrate good performance of the proposed procedure even on non-Gaussian data in a rather challenging regime.