## 5 Multi-Source Fusion Bayesian Compressed Sampling (MSF-BCS)
### Multi-Source Bayesian Compressive Sampling (MS-BCS)
[33; 35] identified and estimated \(\omega_{t}^{\rm 2D,m}\)from the sparse measured data \(\mathbf{Y}^{m}\) in Bayesian framework, allowing \(\mathbf{F}^{m}\) to be properly reconstructed (\(\hat{\mathbf{F}}^{m}=\hat{\mathbf{B}}^{m}\hat{\mathbf{\sigma}}^{2\text{D},m}\)) with the approximation \(\hat{\mathbf{\omega}}^{2\text{D},m}=\left[\hat{\omega}_{1}^{2\text{D},m},\hat{ \omega}_{2}^{2\text{D},m},\ldots,\hat{\omega}_{N_{t}}^{2\text{D},m}\right]\)  
\[p\left(\hat{\mathbf{\omega}}^{2\text{D},m}\mid\mathbf{y}^{2\text{D},m}\right)=K^{m}p \left(\mathbf{y}^{2\text{D},m}\mid\hat{\mathbf{\omega}}^{2\text{D},m}\right)p\left( \hat{\mathbf{\omega}}^{2\text{D},m}\right) \tag{14}\]  
where \(p\left(\mathbf{y}^{2\text{D},m}\mid\hat{\mathbf{\omega}}^{2\text{D},m}\right)\) is the likelihood function, reflecting the plausibility of observing \(\mathbf{y}^{2\text{D},m}\) given \(\hat{\mathbf{\omega}}^{2\text{D},m}\); \(p\left(\hat{\mathbf{\omega}}^{2\text{D},m}\right)\) is the prior probability density function (PDF) of \(\hat{\mathbf{\omega}}^{2\text{D},m}\), reflecting the prior knowledge in the absence of \(\mathbf{y}^{2\text{D},m}\); \(K^{m}\) is a normalizing constant that ensures that the integration of the posterior distribution is unity. According to the analytical posterior PDF description of \(\hat{\mathbf{\omega}}^{2\text{D},m}\) conditioned on variables \(\mathbf{\alpha}^{m}\), \(\tau\), and \(\gamma\) by [PERSON] et al. (2015), the \(\hat{\mathbf{\omega}}^{2\text{D},m}\) posterior PDF follows a multivariate Gaussian distribution, which can be expressed as  
\[p\left(\hat{\mathbf{\omega}}^{2\text{D},m},\mathbf{\alpha}^{m},\tau,\gamma\mid\mathbf{y}^{ 2\text{D},m}\right)=K^{m}p\left(\mathbf{y}^{2\text{D},m}\mid\hat{\mathbf{\omega}}^{2 \text{D},m},\tau\right)p\left(\hat{\mathbf{\omega}}^{2\text{D},m},\mathbf{\alpha}^{m },\tau,\gamma\right) \tag{15}\]  
the mean and covariance matrices of \(\hat{\mathbf{\omega}}^{2\text{D},m}\) are:  
\[ \begin{split}\mathbf{\mu}_{\hat{\mathbf{\omega}}^{2\text{D},m}}=\mathbf{ COV}_{\hat{\mathbf{\omega}}^{2\text{D},m}}\left(\mathbf{A}^{m}\right)^{\text{T}}\mathbf{y}^{2 \text{D},m}\tau\\ \mathbf{COV}_{\hat{\mathbf{\omega}}^{2\text{D},m}}=\left[(\mathbf{A}^{m})^ {\text{T}}\mathbf{A}^{m}\tau+\mathbf{D}^{\mathbf{\alpha},m}\right]^{-1}\end{split}  \tag{16}\]  
where \(\mathbf{y}^{2\text{D},m}=\text{vec}\left[(\mathbf{Y}^{m})^{\text{T}}\right]=A^{m}\hat {\mathbf{\omega}}^{2\text{D},m}\), \(\mathbf{A}^{m}=B_{1}^{\text{D},m}\otimes\mathbf{A}_{2}^{m}\), \(\otimes\) represents the Kronecker product in linear algebra. \(\mathbf{D}^{\mathbf{\alpha},m}\) is an \(N_{t}\times N_{t}\) matrix with diagonal elements \(\mathbf{a}_{t}^{m}\) (t = 1, 2, \(\cdots\), \(N_{t}\)), \(\tau\) follow a Gamma distribution:  
\[p\left(\tau\mid\hat{\mathbf{\omega}}^{2\text{D},m},\mathbf{\alpha}^{m},\gamma,\mathbf{y}^ {2\text{D},m}\right)=d_{n}^{c_{n}}\tau^{c_{n}-1}\exp(-d_{n}\tau)/\Gamma(c_{n}) \tag{17}\]  
where \(c_{n}=\left(N_{x_{1}}^{m}\times M^{m}\right)/2+c_{0}\), \(c_{0}=1\). Moreover, \(d_{n}=d_{0}+1/2[\left(\mathbf{y}^{2\text{D},m}\right)^{\text{T}}\mathbf{y}^{2\text{D},m}-2\left(\hat{\mathbf{\omega}}^{2\text{D},m}\right)^{\text{T}}\mathbf{A}^{m}]^{ \text{T}}\mathbf{y}^{2\text{D},m}+\left(\hat{\mathbf{\omega}}^{2\text{D},m}\right)^{ \text{T}}\mathbf{(}\mathbf{A}^{m}\mathbf{)}^{\text{T}}\mathbf{A}^{m}\hat{\mathbf{\omega}}^{2\text{D },m}\), \(d_{0}=10^{-4}\). It is worth noting that \(\gamma\) also follows the Gamma distribution, \(p(\gamma\mid\hat{\mathbf{\omega}}^{2\text{D},m},\mathbf{\alpha}^{m},\tau,\mathbf{y}^{2 \text{D},m})=(\gamma_{b})^{\gamma_{a}}\gamma^{\gamma_{a}-1}\exp(-\gamma_{b} \gamma)/\Gamma(\gamma_{a})\), where \(\gamma_{a}=N_{t}^{m}+a_{0}\) (let \(a_{0}=10^{-4}\)) and \(\gamma_{b}=b_{0}+\sum\limits_{t=1}^{N_{t}^{m}}(\alpha_{t}^{m})^{-1}\) (let \(b_{0}=10^{-4}\)). \(\mathbf{\alpha}^{m}=\left[\mathbf{a}_{1}^{m},\mathbf{\alpha}_{2}^{m},\ldots,\mathbf{\alpha}_{N_ {t}}^{m}\right]\) follows a generalized inverse Gaussian distribution, which is represented as ([PERSON], 2015):  
\[