## 2 Theoretical Description  
### Model Derivation  
In the time domain, a bearing fault signal corrupted by white Gaussian noise can be represented as  
\[g=x+n_{t}, \tag{1}\]  
where \(g\in R^{N}\) denotes the observed noisy bearing fault signal, \(x\in R^{N}\) denotes the noise-free transient components and \(n_{t}\in R^{N}\) denotes noise in time domain. It is necessary to estimate the transient components \(x\) as accurately as possible based on the noisy signal \(g\) and their characteristics. In order to make the most of the statistical distribution characteristics of related signals, a linear transformation is needed. Wavelet transform is an efficient linear transformation and it transforms a signal in time domain into time-frequency domain, which can better observe the local characteristics of the signal and simultaneously observe the time and frequency information of the signal [26]. At the same time, the computational complexity of wavelet transform is also low, which can ensure the fast extraction of fault features. When the wavelet transform with orthogonality is performed, the model can be represented as  
\[y=w+n, \tag{2}\]  
where \(y\) denotes the wavelet coefficients of the observed noisy bearing fault signal \(g\), \(w\) denotes the wavelet coefficients of the noise-free transient components \(x\) and \(n\) denotes noise. Through the analysis of bearing fault mechanism and statistical analysis of simulation signals, we found that the pdfs of the noise-free transient components and its wavelet coefficients are mixture of two Laplace pdfs [27]:  
\[p(w)=k_{1}\text{lap}(\sigma_{1})+k_{2}\text{lap}(\sigma_{2}), \tag{3}\]  
where  
\[\text{lap}(\sigma)=\frac{1}{\sqrt{2}\sigma}e^{-\frac{\sqrt{2}}{\sigma}|x|}, \tag{4}\]  
and \(k_{1}+k_{2}=1\). And noise follows Gaussian distribution and is independent of transient components [28]. In order to minimize the noise contained in the estimated components \(w\), the MAP estimator is adopted to estimate \(w\) according to noisy wavelet coefficients \(y\)[23]. The MAP estimator for \(w\) in Equation (2) is defined as  
\[\hat{w}(y)=\arg\max_{w}p(w|y). \tag{5}\]  
Based on the [PERSON] theorem, the maximum a posteriori estimator can be expressed as  
\[\hat{w}(y)=\arg\max_{w}\frac{p(y|w)p(w)}{p(y)}. \tag{6}\]As the pdf \(p(y)\) on the denominator is irrelevant with \(w\), the MAP estimator of \(w\) is transformed into the following form  
\[\hat{w}(y)=\arg\max_{w}p(y|w)\cdot p(w). \tag{7}\]  
The Equation (7) is equivalent to solving the following formula  
\[\hat{w}(y)=\arg\max_{w}\ln[p(y|w)\cdot p(w)]. \tag{8}\]  
Consequently, the MAP estimate of \(w\) can be obtained by solving the following formula  
\[\frac{\partial}{\partial w}\ln p(y|w)+\frac{\partial}{\partial w}\ln p(w)=0. \tag{9}\]  
According to the form of the above formula, it is necessary to know the conditional pdf \(p(y|w)\). As for the model in Equation (2), \(p(w|y)\) denotes the pdf of \(n\) when the parameter \(w\) is given  
\[p(y|w)=\frac{1}{\sqrt{2\pi}\alpha_{n}}e^{-\frac{(y-w)^{2}}{2\alpha_{n}^{2}}}, \tag{10}\]  
where \(\alpha_{n}^{2}\) represents the variance of the white zero-mean Gaussian noise. The standard variance of the independent noise \(\alpha_{n}\) also needs to be estimated. The parameter can be estimated through a frequently used method when implementing denoising in wavelet domain  
\[y=w+n, \tag{2}\]0  
where MAD denotes median absolute deviation  
\[y=w+n, \tag{2}\]1  
and \(y_{n}\) denotes the wavelet coefficients in the finest level [29]. In this paper, the estimation is carried out at the first level which which is often the finest level used for estimation in wavelet denoising problems and thus \(y_{n}\) denotes the level 1 detail coefficients. The single Laplace model will be discussed firstly and then extended to the mixture model.