To reduce the economic losses caused by bearing failures and prevent safety accidents, it is necessary to develop an effective method to predict the remaining useful life (RUL) of the rolling bearing. However, the degradation inside the bearing is difficult to monitor in real-time. Meanwhile, external uncertainties significantly impact bearing degradation. Therefore, this paper proposes a new bearing RUL prediction method based on long-short term memory (LSTM) with uncertainty quantification. First, a fusion metric related to runtime (or degradation) is proposed to reflect the latent degradation process. Then, an improved dropout method based on nonparametric kernel density is developed to improve estimation accuracy of RUL. The PHM2012 dataset is adopted to verify the proposed method, and comparison results illustrate that the proposed prediction model can accurately obtain the point estimation and probability distribution of the bearing RUL.