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A crystal oscillator is a typical frequency generating unit that is widely used in computers, neural chips, biosensors and other applications; thus, it is very important to estimate and predict its remaining useful life (RUL) precisely. However, there are few existing RUL prediction methods because the observed data involve various uncertainties, leading to the great limitation of RUL prediction in practical application. In this work, we propose an uncertainty RUL prediction method based on the exponential stochastic degradation model that considers the multiple uncertainty sources of oscillator stochastic degradation processes simultaneously. Next, based on Bayesian theory, a novel Bayesian-Extreme Learning Machine parameter-updating algorithm that combines the local and global similarity methods is presented and used to eliminate the effects of multiple uncertainty sources and predict the RUL accurately. The effectiveness of the method is demonstrated using the accelerated degradation tests of crystal oscillators. Through comparisons with the predicted results without uncertainty, the proposed method demonstrates its superiority in describing the stochastic degradation processes and predicting the oscillator's RUL.