## 4 Bayesian Identification
### Hyperparameter Optimization
The elaborated variant of the cost function in Equation (27) allows reducing the effective number of parameters to be optimized; therefore, this helps to solve the non-convex optimization problem. Note that \(\underline{\mathbf{\Sigma}}\) is a \(D\times D\) matrix, and its inversion has a computational complexity \(O(D)\). The straightforward computation of \(V_{1}(\boldsymbol{\theta})\) can therefore be quite time consuming as \(F\) is on the order of hundreds or even thousands for a vibration test performed with the DIC technique. The numerical accuracy is another issue since both \(\underline{\mathbf{\Sigma}}\) and \(\boldsymbol{\
abla}\boldsymbol{\
abla}^{\mathrm{H}}\) might be ill conditioned, and they can possibly have a much larger magnitude than \(\mathbf{P}\). The computation of \(h(\boldsymbol{\theta})\) and \(V_{0}(\boldsymbol{\theta})\) is carried out by implementing an efficient and accurate numerical algorithm [26]. The numerical procedure starts by finding the Cholesky decomposition of \(\underline{\mathbf{\Sigma}}_{\mathrm{H}}\)  
\[\underline{\mathbf{\Sigma}}_{\mathrm{H}}=\mathbf{L}\mathbf{L}^{\mathrm{H}} \tag{29}\]  
with \(\mathbf{L}\) being the lower triangular matrix. Using Equation (29), [PERSON]'s determinant theorem, and the Woodbury matrix inversion lemma [27], it is derived that  
\[\ln\det\underline{\mathbf{\Sigma}}=\ln\det\left(\mathbf{L}^{\mathrm{H}} \boldsymbol{\Psi}^{\mathrm{H}}\underline{\mathbf{P}}^{-1}\boldsymbol{\Psi} \mathbf{L}+\mathbf{I}_{W}\right)+\ln\det\underline{\mathbf{P}} \tag{30}\]  
\[\underline{\mathbf{Y}}^{\mathrm{H}}\underline{\mathbf{\Sigma}}^{-1}\underline {\mathbf{Y}}=\underline{\mathbf{Y}}^{\mathrm{H}}\underline{\mathbf{P}}^{-1} \underline{\mathbf{Y}}-\underline{\mathbf{Y}}^{\mathrm{H}}\underline{\mathbf{ P}}^{-1}\boldsymbol{\Psi}\mathbf{L}\left(\mathbf{L}^{\mathrm{H}}\boldsymbol{\Psi}^{ \mathrm{H}}\underline{\mathbf{P}}^{-1}\boldsymbol{\Psi}\mathbf{L}+\mathbf{I} _{W}\right)^{-1}\mathbf{L}^{\mathrm{H}}\boldsymbol{\Psi}^{\mathrm{H}} \underline{\mathbf{P}}^{-1}\underline{\mathbf{Y}} \tag{31}\]where \(\mathbf{I}_{W}\) is a \(W\times W\) identity matrix. Note that the determinant calculation in Equation (30) and matrix inversion in Equation (31) are independent of the number of data. Moreover, a thin QR decomposition is performed,  
\[ \begin{bmatrix}\mathbf{E}^{-\frac{1}{2}}\mathbf{\Psi}\mathbf{L}&\mathbf{E}^{- \frac{1}{2}}\mathbf{\chi}\\ \mathbf{I}_{W}&\mathbf{0}\end{bmatrix} =\mathbf{Q} \begin{bmatrix}\mathbf{R}_{ 1}&\mathbf{R}_{2}\\ \mathbf{0}&R_{3}\end{bmatrix}  \tag{32}\]  
where \(\mathbf{Q}\) is an orthogonal matrix, \(\mathbf{R}_{1}\) is a \(W\)-dimensional upper triangular matrix, \(\mathbf{R}_{2}\) is a column vector of length \(W\), and \(R_{3}\) is a scalar. By virtue of such QR decomposition,  
\[\mathbf{L}^{H}\mathbf{\Psi}^{H}\mathbf{\underline{P}}^{-1}\mathbf{\Psi} \mathbf{L}+\mathbf{I}_{W}=\mathbf{R}_{1}^{H}\mathbf{R}_{1} \tag{33}\]  
\[\mathbf{L}^{H}\mathbf{\Psi}^{H}\mathbf{\underline{P}}^{-1}\mathbf{\chi}= \mathbf{R}_{1}^{H}\mathbf{R}_{2} \tag{34}\]  
\[\mathbf{\chi}^{H}\mathbf{\underline{P}}^{-1}\mathbf{\chi}=\mathbf{R}_{2}^{H} \mathbf{R}_{2}+R_{3}^{2} \tag{35}\]  
Therefore, \(h(\mathbf{\theta})\) and \(V_{0}(\mathbf{\theta})\) are obtained in the following way:  
\[ \begin{array}{l}h(\mathbf{\theta})=\exp(2\text{Re}(\ln\text{det}\mathbf{R}_{1}) /D)\\ V_{0}(\mathbf{\theta})=R_{3}^{2}/D\end{array}  \tag{36}\]  
where \"Re\" takes the real part of a complex variable. This allows the cost function \(V_{1}(\mathbf{\theta})\) to be efficiently and accurately calculated. The gradient-based optimization algorithm is applied to minimize the cost function \(V_{1}(\mathbf{\theta})\).