## 1 Introduction
ML models have been shown to facilitate a high level of prediction accuracy, while the performance may not be guaranteed without a substantial volume of full lifetime data, which is often difficult to obtain. Alternatively, Bayesian statistical approaches are another appealing branch within the realm of data-driven models given their superiority in integrating prior or domain information; they are free of reliance on massive quantities of historical data and also interpretable with uncertainty quantification. The RUL of a bearing can be viewed as a random variable depending on the current health conditions observed from the CM measurements. In this sense, the probability density of the RUL conditioned on historical observations becomes the main interest of the statistical model [3], where Bayesian inference is often adopted to obtain the posterior estimates of the RUL. Recently, stochastic processes [14; 15] and recursive Bayesian filters [16; 17] have demonstrated promising performance in RUL estimation and offer a flexible framework for incorporating prior knowledge and updating beliefs based on new observations. The Kalman filter (KF) and its variants [18; 19], the extended Kalman filter (EKF) [20] and unscented Kalman filter (UKF) [21], are efficient state estimation tools for dynamical systems given inaccurate models and noisy observations and have been frequently utilized in various industrial applications, such as the tracking of satellite trajectories [22] and the positioning of autonomous vehicles [23]. In the case of bearing condition monitoring, the latent degradation level is the target of state estimation behind the observable CM measurements, such as vibration acceleration signals, or the bearing health indicators (HI) obtained from certain feature engineering techniques. For example, [PERSON] et al. established the Wiener process-based exponential model for bearing degradation and applied the KF for removing random noise in the RUL prediction results [15]. According to [19], the KF is endowed with the capability of providing the optimal state estimation under the Gaussian noise assumption, which minimizes the mean square error between the true and the estimated state. Still, the effectiveness of KF can be limited due to the linear assumption of the system dynamics. The EKF generalizes the KF to handle nonlinear system dynamics by introducing local linearization based on Taylor's expansion. Considering the complexity of the bearing degradation process, [PERSON] et al. applied EKF to predict the RUL of bearings based on the entropy level of the time-frequency distribution and the root-mean-square (RMS) value of vibration signals [16]. Two distinct exponential state-space models are hypothesized and compared to represent the trajectories of the health indicators and estimate the true health state. The applications to the PROGNOSTIA data also reveal the efficiency of EKF for state estimation, where convergent prediction of RUL is achieved as the bearing approaches the end of its lifespan [16]. Despite notable progress, some challenges persist in the realm of bearing RUL prediction. As with the importance of feature extraction to robust fault diagnosis [24; 25], feature extraction is one crucial step that affects the reliability of the RUL prediction. Conventional bearing health indicators based on statistical measures of vibration signals may encounter interference components arising from synchronous speed vibrations of the connecting structure, such as a faulty rotor with misalignment. These interfering components could greatly undermine the robustness of an RUL prediction. In addition, despite the flexibility of the Bayesian filters, their sensitivity to prior assumptions and initial states often lead to a large estimation error within the early predictions of RUL, which is less favoured in engineering practice and should be further improved.