Electronic Journal of Statistics

Bayesian inference for spectral projectors of the covariance matrix

Igor Silin and Vladimir Spokoiny

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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.

Article information

Source
Electron. J. Statist., Volume 12, Number 1 (2018), 1948-1987.

Dates
Received: March 2018
First available in Project Euclid: 18 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1529308884

Digital Object Identifier
doi:10.1214/18-EJS1451

Mathematical Reviews number (MathSciNet)
MR3815302

Zentralblatt MATH identifier
06917428

Subjects
Primary: 62F15: Bayesian inference 62H25: Factor analysis and principal components; correspondence analysis 62G20: Asymptotic properties
Secondary: 62F25: Tolerance and confidence regions

Keywords
Covariance matrix spectral projector principal component analysis Bernstein–von Mises theorem

Rights
Creative Commons Attribution 4.0 International License.

Citation

Silin, Igor; Spokoiny, Vladimir. Bayesian inference for spectral projectors of the covariance matrix. Electron. J. Statist. 12 (2018), no. 1, 1948--1987. doi:10.1214/18-EJS1451. https://projecteuclid.org/euclid.ejs/1529308884


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