Bayesian inference for spectral projectors of the covariance matrix

Igor Silin, Vladimir Spokoiny

    Research output: Contribution to journalArticlepeer-review

    3 Citations (Scopus)

    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 PJ. Recent paper [24] investigated the asymptotic distribution of the Frobenius distance between the projectors ‖ PJ − P J2, 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 Σ in a vicinity of Σ. Numerical simulations illustrate good performance of the proposed procedure even on non-Gaussian data in a rather challenging regime.

    Original languageEnglish
    Pages (from-to)1948-1987
    Number of pages40
    JournalElectronic Journal of Statistics
    Volume12
    Issue number1
    DOIs
    Publication statusPublished - 2018

    Keywords

    • Bernstein
    • Covariance matrix
    • Principal component analysis
    • Spectral projector
    • Von mises theorem

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