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Eigenvector-Assisted Statistical Inference for Signal-Plus-Noise Matrix Models

In this paper, we develop a generalized Bayesian inference framework for a collection of signal-plus-noise matrix models arising in high-dimensional statistics and many applications. The framework is built upon an asymptotically unbiased estimating equation with the assistance of the leading eigenvectors of the data matrix. The solution to the estimating equation coincides with the maximizer of an appropriate statistical criterion function. The generalized posterior distribution is constructed by replacing the usual log-likelihood function in the Bayes formula with the criterion function. The proposed framework does not require the complete specification of the sampling distribution and is convenient for uncertainty quantification via a Markov Chain Monte Carlo sampler, circumventing the inconvenience of resampling the data matrix. Under mild regularity conditions, we establish the large sample properties of the estimating equation estimator and the generalized posterior distributions. In particular, the generalized posterior credible sets have the correct frequentist nominal coverage probability provided that the so-called generalized information equality holds. The validity and usefulness of the proposed framework are demonstrated through the analysis of synthetic datasets and the real-world ENZYMES network datasets.