  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.
