## 1 Introduction  
Rolling bearings are key parts of much of the modern mechanical equipment which plays an essential role in social production and living, such as engines, generators, machine tools, and so on. As rolling bearings usually operate under harsh working conditions and need to work long hours, they are prone to failures [1]. So, it is crucial to diagnose faults of the rolling bearings as soon as possible to avoid safety accidents and maintain good working efficiency. Vibration signals generated by faulty bearings directly reflect the dynamic characteristics of faulty bearings [2]. Therefore, it is effective and reasonable to extract the features of the repetitive transient components by analyzing the vibration signals of faulty bearings. However, the vibration signals measured are always drowned out by strong noise, which makes the features of the repetitive transient component difficult to extract [3]. Therefore, methods to effectively extract the features of the repetitive transient component from the noisy vibration signals are key to bearing fault diagnosis. To deal with this issue, various effective methods, such as time-frequency analysis [4], spectral kurtosis [5], empirical mode decomposition [6], wavelet analysis [7], stochastic resonance [8], deep learning [9], sparse representation [10] and bispectrum analysis [11] have been proposed over the past decade. The time-frequency analysis method with high time-frequency aggregation has achieved good results in the extraction of frequency conversion curve and the extraction of fault feature frequency [12]. The empirical mode decomposition method has good time-frequency resolution and adaptability and is able to separate the noise and effective component by reasonably selecting the intrinsic mode functions without giving a given transform base [13]. The spectral kurtosis filtering method, based on the non-Gaussian property exhibited by the fault component, designs the filter to select the resonant frequency band of the fault component according to the principle of maximizing the statistic such as kurtosis, and then determines the fault feature frequency according to envelope spectrum analysis [14]. The deep learning method can adaptively extract features based on the model constructed, which provides a new idea for improving the more accurate and effective fault diagnosis of rolling bearings [15]. Sparse representation has been successfully applied to mechanical fault diagnosis and achieved significant results. As an optimization method, the sparse representation method needs to optimize an objective function, which consists of the fidelity term, penalty term and regularization parameter [16]. In the field of mechanical fault diagnosis, research on sparse representation has focused on the construction of penalty functions [17]. At the same time, in the sparse representation method an over-complete dictionary according to the characteristics of the components needs to be constructed to be estimated in advance [18]. Modulation signal bispectrum (MSB) is a representative of bispectrum analysis-based methods. The MSB method has inherent noise suppression capabilities and can help enhance the modulation effects of a bearing fault and achieve the goal of frequency band optimization for fault detection [19]. The aforementioned methods have achieved good results in fault feature extraction. However, most methods described above cannot satisfy the requirement of timeliness very well. From the point of view of the timeliness of fault feature extraction, a method to quickly estimate the transient components of noisy bearing fault signals and extract fault features is needed. Many methods based on probability and statistics have been proposed and successfully applied to many fields, including fault diagnosis and detection [20]. These methods can mainly be divided into three categories: maximum-likelihood (ML) [21], Bayesian [22], and maximum-a-posteriori (MAP) approaches [23].