## Appendix B  
Lemma 1 denotes that \(\sigma_{m}\)\({}^{-q}(\widetilde{w})\) is maximized at \(\widetilde{w}_{0}\). By (A5), we have,  
\[\text{grad}\left\{\ln[p_{m}\text{${}^{-q}(\widetilde{w})$}]\right\}|_{ \widetilde{w}_{0}}=-\frac{q}{p_{m}(\widetilde{w})}\left[\left.\left\{\frac{ \partial\sigma_{m}(\widetilde{w})}{\partial[\widetilde{w}]_{j}}\right\} \right|_{I=1,2,\ldots,2N_{\mathbb{R}}}\right]\right|_{\widetilde{w}=\widetilde {w}_{0}}=0\] (A11)  
By (A5), (A6), and (A10), the Taylor series of \(\ln[\sigma_{m}\text{${}^{-q}(\widetilde{w})$}]\) is expanded to  
\[\ln[\sigma_{m}\text{${}^{-q}(\widetilde{w})$}]\approx\ln[\sigma_{m}\text{${}^{- q}(\widetilde{w}_{0})$}]+\frac{1}{2}(\widetilde{w}-\widetilde{w}_{0})^{ \mathsf{T}}\mathbf{H}_{\widetilde{w}_{0}}(\widetilde{w}-\widetilde{w}_{0})\] (A12)  
Note that a positive definite matrix \(\mathbf{A}\) is invertible, and the determinant of \(\mathbf{A}\) can be computed by \(\exp[\text{Tr}\ln(\mathbf{A})]\), where \(\text{Tr}\mathbf{A}\) stands for the trace of \(\mathbf{A}\). Recalling (A1) and (A6), the saddle point approximation can be computed by Gaussian integral  
\[\begin{array}{l}\int_{\mathbf{w}}\frac{p_{m}\text{${}^{-q}(\mathbf{w})$}}{\pi^{N_{ \mathbb{R}}}\sigma^{2}\text{${}^{\mathbb{R}}$}}\text{d}\mathbf{w}&\approx\frac{p_{m }\text{${}^{-q}(\widetilde{w}_{0})$}}{\pi^{N_{\mathbb{R}}}p_{m}\text{${}^{ \mathbb{R}}$}}\int_{\widetilde{w}}\exp\left(\frac{1}{2}(\widetilde{w}- \widetilde{w}_{0})^{\mathsf{T}}\mathbf{H}_{\widetilde{w}_{0}}(\widetilde{w}- \widetilde{w}_{0})\right)\text{d}\widetilde{w}\\ &=\left\{p_{m}(\widetilde{w}_{0})\text{det}^{1/2q}\Big{(}-\frac{\sigma^{2}}{2} \mathbf{H}_{\widetilde{w}_{0}}\Big{)}\right\}^{-q}\geq\left\{p_{m}( \widetilde{w}_{0})*\frac{1}{2q}\text{Tr}\Big{(}-\frac{\sigma^{2}}{2}\mathbf{H} _{\widetilde{w}_{0}}\Big{)}\right\}^{-q}\\ &\geq\sum\limits_{k=1}^{N}\exp\left\{\begin{array}{c}-\frac{\|c_{m,k}\|^{2} }{\|c_{m,k}\|^{2}}}\\ 1-q\frac{\|\widetilde{c}_{m,k}\|^{2}}{\|c_{m,k}\|^{2}}+q\frac{c_{m,k}\|^{2} \epsilon_{m,k}+c_{m,k}^{2}\epsilon_{m,k}}{\|c_{m,k}\|^{2}}-\left(\frac{\|c_{m,k} \|^{2}}{\sigma^{2}}\right)^{-1}\\ \ln\left(\sum\limits_{k=1}^{N}\frac{[\mathbf{w}_{m}\|c_{m,k}\|^{2}}{\sigma^{2}}- \frac{\|c_{m}\|^{2}}{\sigma^{2}}\right)^{-1}\end{array}\right]\end{array}\right\} ^{-q}\] (A13)According to (A5), we can induce  
\[\sum_{k=1}^{N}\left(\overset{-}{c}_{m}-c_{m,k}\right)\exp[-\frac{||\epsilon_{m,k} ||^{2}}{\sigma^{2}}\left(1+q\frac{c_{m,k}\mathrm{H}^{-}\overset{-}{c}_{m}+ \overset{-}{c}_{m}\mathrm{H}c_{m,k}}{\sigma^{2}}-q\frac{||\overset{-}{c}_{m}|| ^{2}}{\sigma^{2}}\right)]=0 \tag{14}\]  
Therefore, (A13) and (A14) demonstrate that Gaussian integral is dominated by \(||c_{m,k}||^{2}/\sigma^{2}\) in terms of exponential. Define a multiplier \(\alpha_{m,k}\) dominated by \(||c_{m,k}||^{2}/\sigma^{2}\),  
\[\alpha_{m,k}=\left[\begin{array}{c}1-q\frac{||\overset{-}{c}_{m}||^{2}}{|| \epsilon_{m,k}||^{2}}+q\frac{c_{m,k}\mathrm{H}^{-}\overset{-}{c}_{m}+\overset {-}{c}_{m}\mathrm{H}c_{m,k}}{||\epsilon_{m,k}||^{2}}-\left(\frac{||c_{m,k}||^{ 2}}{\sigma^{2}}\right)^{-1}\\ \ln(\sum\limits_{k=1}^{N}\frac{|\mathbf{\rho}_{m,k}||_{m,k}||^{2}}{\sigma^{2}}- \frac{||\overset{-}{c}_{m}||^{2}}{\sigma^{2}})\end{array}\right] \tag{15}\]  
So  
\[\int_{\mathbf{w}}\frac{1}{\pi^{N_{\text{R}}}\sigma^{2N_{\text{R}}}}\left(\sum \limits_{k=1}^{N}p_{m,k}(\mathbf{w})\right)^{-q}\mathrm{d}\mathbf{w}\approx\left[ \sum\limits_{k=1}^{N}\exp(-\frac{\alpha_{m,k}||\mathbf{c}_{m,k}||^{2}}{\sigma^{2}} )\right]^{-q} \tag{16}\]