# Bayesian Compressive Sensing Based Optimized Node Selection Scheme in Underwater Sensor Networks
## 2 System Model and Problem Formulation
### Bayesian Compressive Sensing  
In this subsection, the sparse vector \(\mathbf{s}\) is reconstructed by measurement \(\mathbf{y}\) according to Bayesian estimation. Compared with the traditional compressive sensing reconstruction algorithm, the advantage of Bayesian compressive sensing is that the noise variance can also be estimated. In addition, the posterior probability density function can be given through some prior information. The main idea of Bayesian estimation is taking full advantage of reasonably assumed prior information and the prior information including the distribution of noise and the measurement \(\mathbf{y}\). Firstly, the Gaussian likelihood model of the measurement \(\mathbf{y}\) can be written as follows [28]:  
\[p(\mathbf{y}|\mathbf{s},\sigma^{  
\[p(\mathbf{s}|\boldsymbol{\alpha}) =\prod_{i=1}^{N}\mathcal{N}(s_{i}|0,\alpha_{i}^{-1}) \tag{8}\] \[=\prod_{i=1}^{N}(2\pi)^{-\frac{1}{2}}\alpha_{i}^{\frac{1}{2}}e^{- \frac{\alpha_{i}^{2}\sigma_{i}^{2}}{2}},\]  
where \(\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{N})^{T}\) is the hyperparameter. Hierarchical priors about the hyperparameters \(\boldsymbol{\alpha}\), \(\sigma^{2}\) and the sparse vector \(\mathbf{s}\) are given in Appendix A. It is conjugate with Gaussian distribution. The marginal likelihood function \(p(\mathbf{y}|\boldsymbol{\alpha},\sigma^{2})\) can be computed by integrating \(p(\mathbf{y}|\mathbf{s},\sigma^{2})p(\mathbf{s}|\boldsymbol{\alpha})\)  
\[p(\mathbf{y}|\boldsymbol{\alpha},\sigma^{2}) =\int p(\mathbf{y}|\mathbf{s},\sigma^{2})p(\mathbf{s}|\boldsymbol {\alpha})d\mathbf{s} \tag{9}\] \[=(2\pi)^{-\frac{M}{2}}|\boldsymbol{\Sigma}|^{-\frac{1}{2}}e^{- \frac{\mathbf{y}^{T}\mathbf{C}^{-1}\mathbf{y}}{2}},\]  
where \(\mathbf{C}=\sigma^{2}\mathbf{I}+\mathbf{A}\boldsymbol{\Lambda}^{-1}\mathbf{ A}^{T}\) and \(\boldsymbol{\Lambda}\) is a diagonal matrix of the hyperparameters \(\boldsymbol{\alpha}\). According to the assumption of the hierarchical prior and the Gaussian likelihood model, the closed form solution of the posterior distribution can be derived:  
\[p(\mathbf{s}|\boldsymbol{\alpha},\sigma^{2},\mathbf{y}), \tag{10}\] \[= (2\pi)^{-\frac{N}{2}}|\boldsymbol{\Sigma}|^{-\frac{1}{2}}e^{- \frac{(\mathbf{s}-\boldsymbol{\mu})^{T}\mathbf{C}^{-1}(\mathbf{s}-\boldsymbol{ \mu})}{2}},\]  
where the mean and covariance matrix are \(\boldsymbol{\mu}\) and \(\boldsymbol{\Sigma}\) as follows:  
\[\boldsymbol{\mu} =\sigma^{-2}\boldsymbol{\Sigma}\boldsymbol{\Lambda}^{T}\mathbf{y} \tag{11}\] \[\boldsymbol{\Sigma} =(\sigma^{-2}\boldsymbol{\Lambda}^{T}\boldsymbol{\Lambda}+ \boldsymbol{\Lambda})^{-1}.\]  
It can be seen that \(\boldsymbol{\mu}\) and \(\boldsymbol{\Sigma}\) are functions of the hyperparameters \(\boldsymbol{\alpha}\) and \(\sigma^{2}\) which are unknown. It is necessary to estimate them to obtain the mean and variance of the sparse vector \(\mathbf{s}\). The hyperparameters' estimation can be obtained by maximizing \(p(\mathbf{y}|\boldsymbol{\alpha},\sigma^{2})p(\boldsymbol{\alpha})p(\sigma^{ -2})\). Due to a uniform distribution assumption of \(\boldsymbol{\alpha}\) and \(\sigma^{2}\), the hyperparameters' estimation can be obtained by maximizing the marginal likelihood function \(p(\mathbf{y}|\boldsymbol{\alpha},\sigma^{2})\) directly, or equivalently maximizing the logarithm of \(p(\mathbf{y}|\boldsymbol{\alpha}:\sigma^{2})\). \[\mathcal{L} =\text{log}p(\mathbf{y}|\boldsymbol{\alpha},\sigma^{2}) \tag{12}\] \[=-\frac{1}{2}(M\text{log}(2\pi)+\text{log}|\mathbf{C}|+\mathbf{y} ^{T}\mathbf{C}^{-1}\mathbf{y}).\]  
By taking the derivative of \(\mathcal{L}\) about \(\boldsymbol{\alpha}\) and setting it as zero, the estimation of \(\boldsymbol{\alpha}\) can be given as follows:  
\[=\prod_{i=1}^{N}(2\pi)^{-\frac{1}{2}}\alpha_{i}^{\frac{1}{2}}e^{- \frac{\alpha_{i}^{2}\sigma_{i}^{2}}{2}},\]0  
where \(\Sigma_{ii}\) is the \(i\)th element of the covariance matrix \(\boldsymbol{\Sigma}\).