## 4 Our Proposed GCI-IGGM-PMBM Scheme
### IGGM Implementation  
The closed-form solution using adaptive newborn distributions accompanied by augmented unknown detection probability can be derived using an inverse gamma Gaussian mixture. The gamma, inverse gamma, and Gaussian distributions are presented as the unknown detection probability, adaptive newborn distribution, and target state distribution, respectively. The representations for gamma and inverse gamma distributions were described in Section 3.4, and the recursion of IGGM-PMBM is proposed below. It is assumed that the IGG component at time \(k-1\) is given as  
\[p_{k|k}(X,\xi)=\mathcal{GAM}(\xi_{\xi,i};\alpha_{\xi,k-1},\beta_{\xi,k-1}) \times\mathcal{IG}(X,\xi_{\xi,i};\alpha_{B,k-1},\beta_{B,k-1})\mathcal{N}(X, \xi_{B};m_{B,k-1},P_{B,k-1})\times\mathcal{N}(X;m_{x,k-1},P_{x,k-1}), \tag{56}\]  
\[l_{k}(M|X,\xi)=\mathcal{GAM}(\xi_{\xi,i};\alpha_{\xi,k-1},\beta_{\xi,k-1}) \mathcal{N}(M,\xi;H_{k}X,R_{k}). \tag{57}\]  
The adaptive newborn distribution and target state Gaussian distribution can be simplified as IGG; then, the IGG prediction can be derived as  
\[p_{k|k-1}(X,\xi)=\mathcal{GAM}(\xi_{k|k-1};\alpha_{\xi,k|k-1},\beta_{\xi,k|k-1 })\times\mathcal{IGG}(X,\xi;\alpha_{B,k|k-1},\beta_{B,k|k-1},m_{B,k|k-1},P_{B,k |k-1},m_{X,k|k-1},P_{x,k|k-1}), \tag{58}\]  
\[\text{where }\alpha_{\xi,k|k-1}=\frac{\alpha_{k-1}}{T_{k}},\ \beta_{\xi,k|k-1}=\frac{\beta_{k-1}}{T_{k}},\ \alpha_{B,k|k-1}=\frac{\alpha_{k-1}}{T_{B}},\ \beta_{B,k|k-1}=\frac{\beta_{k-1}}{T_{B}},\ m_{B,k|k-1}=Fm_{B,k-1},\]  
\[\begin{array}{l}P_{B,k|k-1}=FP_{B,k|k-1}+D_{B,k|k-1},\ m_{x,k|k-1}=Fm_{x,k-1}, \text{ and }P_{x,k|k-1}=FP_{x,k-1}F+Q_{x,k|k-1}.\\ T_{\xi}\text{ and }T_{B}\text{ denote the measurement rate parameters for adaptive detection probability and adaptive newborn probability, respectively. \(F,Q_{B,k|k-1},Q_{x,k|k-1}\) denotes the transition matrix, process covariance for newborn distribution, and process covariance for the target state. Then, the IGG updating can be derived as  
\[p_{k}(X,\xi)=\mathcal{GAM}(\xi_{k};\alpha_{\xi,k},\beta_{\xi,k})\times \mathcal{IGG}(X;\alpha_{B,k},\beta_{B,k},m_{B,k},P_{B,k},m_{x,k},P_{x,k}), \tag{59}\]  
where \(\alpha_{\xi,k}\)= \(\alpha_{\xi,k|k-1}+|M_{k}|\), \(\beta_{\xi,k}\)= \(\beta_{\xi,k|k-1}+1\), \(\alpha_{B,k}\)= \(\alpha_{B,k|k-1}+|B_{k}|\), \(\beta_{B,k}\)= \(\beta_{B,k|k-1}+1\), \(m_{B,k}=m_{B,k|k-1}+K\epsilon\), \(P_{B,k}=P_{B,k|k-1}-KHP_{B,k|k-1}\), \(m_{x,k}=m_{x,k|k-1}+K\epsilon\), and \(P_{x,k}=P_{x,k|k-1}-KHP_{x,k|k-1}\). \(H\) denotes the measurement matrix, and \(K,\epsilon\) denote the new observations or measurements based on state transition. The intensity of the newborn target can be assumed to be an IGGM form as follows:  
\[ \begin{array}{ll}D_{k-1}^{B}(X,\xi)&=\sum\limits_{l=1}^{I_{k-1}^{B}}w_{k-1 }^{l,u}\mathcal{IG}(X,\xi_{B};\alpha_{B,k-1}^{l,u},\beta_{B,k-1}^{l,u}) \mathcal{IG}(X;m_{B,k-1}^{l,u},P_{B,k-1}^{l,u})\\ &=\sum\limits_{l=1}^{I_{k-1}^{B}}w_{k-1}^{l,B}\mathcal{IG}(X,\xi_{B};\alpha_{B,k-1}^{l,B},\beta_{B,k-1}^{l,B},m_{B,k-1}^{l,B},P_{B,k-1}^{l,B})\end{array}  \tag{60}\]  
Therefore, the intensity of the Poisson process at time \(k-1\) can also be presented as an IGGM form:  
\[D_{k-1}^{u}(X,\xi)=\sum\limits_{l=1}^{I_{k-1}^{u}}w_{k-1}^{l,u}\mathcal{GAM}( \xi_{\xi,i};\alpha_{\xi,k-1}^{l,u},\beta_{\xi,k-1}^{l,u},\beta_{B,k|k-1}^{l,u},m_{B,k|k-1}^{l,u},P_{B,k|k-1}^{l,u},m_{X,k|k-1}^{l,u},P_{X,k|k-1}^{l,u}). \tag{61}\]  
\[\text{The density of MBM in an IGGM form with parameters }\left\{w_{k-1}^{i}\left\{r_{k-1}^{i},p_{k-1}^{i}(X_{k-1}^{i},x_{k-1}^{i}) \right\}_{i\in\mathbb{I}}\right\}_{j\in\mathbb{J}}\]  
\[\text{can be presented as}\]  
\[p_{k-1}^{i,i}(X,\xi)=\mathcal{GAM}(\xi_{\xi,i};\alpha_{\xi,k-1}^{j,i},\beta_{ \xi,k-1}^{j,i})\mathcal{IGG}(X,\xi_{\xi};\alpha_{B,k-1}^{j,i},\beta_{B,k-1}^{ j,i},m_{B,k-1}^{j,i},P_{B,k-1}^{j,i},m_{X,k-1}^{j,i},P_{X,k-1}^{j,i}). \tag{62}\]  
\[\text{Proposition 3 (Predicted process).