## 2 Preliminary  
### Compressive Sensing  
CS theory proves that if a signal \(\mathbf{z}\) of dimension \(N\) is sufficiently sparse in a certain domain; that is \(\mathbf{z}=\mathbf{\Psi}\mathbf{x}\) and \(\left\|\mathbf{x}\right\|_{0}=K,K\ll N\), where \(\mathbf{\Psi}\in\mathbb{R}^{N\times N}\) is an orthogonal basis and \(\mathbf{x}\in\mathbb{R}^{N}\), it is efficient to recover \(\mathbf{z}\) from a random measurement vector that is obtained by \(\mathbf{y}=\mathbf{\Phi}\mathbf{z}\). \(M\), the length of \(\mathbf{y}\) is much smaller than \(N\). \(\mathbf{\Phi}\) is named as a measurement matrix or sensing matrix. In the sequel, the random non-zero coefficients in \(\mathbf{\Phi}\) obey Gaussian distributions. In general, it is an ill-posed problem to recover signal \(\mathbf{z}\) from \(\mathbf{y}\) since \(\mathbf{y}\) has much smaller dimensions than \(\mathbf{z}\). We have to minimize the solution's \(\ell_{0}\) norm to find the sparsest result. However, it is an NP-hard problem, which is not capable to be represented by mathematical formulas. [PERSON] et al. [PERSON] et al. (2018) state that as long as a measurement matrix satisfies the restricted isometry property (RIP) condition, the \(\ell_{0}\) norm can be replaced by \(\ell_{1}\) norm. A measurement matrix satisfies RIP of order of \(K\) when  
\[\left(1-\epsilon\right)\left\|\mathbf{x}\right\|_{2}\leq\left\|\mathbf{\Phi} \mathbf{\Psi}\mathbf{x}\right\|_{2}\leq\left(1+\epsilon\right)\left\|\mathbf{ x}\right\|_{2} \tag{1}\]  
for all \(\left\|\mathbf{x}\right\|_{0}\leq 2K\) and some \(0<\epsilon<1\). Afterwards, many methods for such a convex optimization problem were introduced to solve the \(\ell_{1}\) norm optimization problem (Rap  
where \(\alpha_{i}\) is the inverse-variance of \(x_{i}\) and \(\alpha\) satisfies a Gamma prior with parameters \(a\) and \(b\) with \(p(\alpha)=\prod_{i=1}^{N}\Gamma(\alpha_{i}|a,b)\). By marginalizing over the hyperparameter \(\alpha\), the overall prior is given by  
\[p(\mathbf{x}|a,b)=\prod_{i=1}^{N}\int_{0}^{\infty}\mathcal{N}(x_{i}|0,\alpha_{ i}^{-1})\Gamma(\alpha_{i}|a,b)da_{i}. \tag{4}\]  
Likewise, the hyperparameter \(\alpha_{0}=1/\sigma^{2}\) satisfies a Gamma prior \(\Gamma(\alpha_{0}|c,d)\). Note that \(a,b,c\), and \(d\) are all set to zero to obtain a uniform hyperprior. The goal of BCS is to estimate three parameters \(\alpha\), \(\alpha_{0}\), and the most important one \(\mathbf{x}\). The posterior function of all unknowns is  
\[p(\mathbf{x},\boldsymbol{\alpha},\alpha_{0}|\mathbf{y})=p(\mathbf{x}| \mathbf{y},\boldsymbol{\alpha},\alpha_{0})p(\boldsymbol{\alpha},\alpha_{0}| \mathbf{y}). \tag{5}\]  
The posterior of \(\mathbf{x}\) is  
\[p(\mathbf{x}|\mathbf{y},\boldsymbol{\alpha},\alpha_{0})=(2\pi)^{-\frac{N}{2}} |\boldsymbol{\Sigma}|^{-\frac{1}{2}}\exp(-\frac{1}{2}\mathbf{x}-\boldsymbol{ \mu}^{T}\boldsymbol{\Sigma}^{-1}\mathbf{x}\boldsymbol{\mu}), \tag{6}\]  
with the mean matrix \(\boldsymbol{\mu}\) and the covariance matrix \(\boldsymbol{\Sigma}\),  
\[\boldsymbol{\mu} =\alpha_{0}\boldsymbol{\Sigma}\boldsymbol{\Theta}^{T}\mathbf{y} \tag{7}\] \[\boldsymbol{\Sigma} =(\alpha_{0}\boldsymbol{\Theta}^{T}\boldsymbol{\Theta}+\mathbf{A} ^{-1})^{-1}.\]  
For the case of uniform hyperpriors,  
\[p(\boldsymbol{\alpha},\alpha_{0}|\mathbf{y})\propto p(\mathbf{y}|\boldsymbol{ \alpha},\alpha_{0}). \tag{8}\]  
Finally, we can estimate all unknown parameters by maximizing the posterior in Equation (5). To find a approximate solution, we use EM algorithm to achieve the solution iteratively.