In this article, we propose an efficient correlated Bayesian inference (CBI) method to estimate the system-level failure rates for large-scale circuit systems over multiple process corners. The key idea is to encode the correlations of circuit performances among the different corners into the prior distributions of several carefully defined failure events. The hyper-parameters of these distributions can be learned from a few simulation samples via Bayesian inference and, next, the system-level failure rates over different corners can be simultaneously estimated by taking into account these prior distributions. An iteratively constrained inference method is further developed to guarantee the numerical stability of the proposed method and legalize all estimated failure rates. The numerical experiments demonstrate that compared to the state-of-the-art algorithm, the proposed method can achieve around 10× runtime reduction without surrendering any accuracy.