3.1.2. GLMB Filter with Objects Spawning

For the prior density which is a GLMB density as in Equation (17) and the transition kernel defined in Equation (6), by applying the joint predict–update approach, a proposal density can be written as follows [30]:

$$\left| \left( \mathfrak{X}\_{+}(\mathbb{X}\_{+}|\mathrm{Z}\_{+}) \propto \mathfrak{A}(\mathbb{X}\_{+}) \sum\_{I\_{\mathfrak{s}} \mathbb{Z}\_{+} \theta\_{+}} \omega^{(I\_{\mathfrak{s}}\overline{\mathfrak{s}}\_{\mathfrak{t}})} \widetilde{\omega}\_{\mathbb{Z}\_{+}}^{(I\_{\mathfrak{s}}\overline{\mathfrak{s}}\_{\mathfrak{t}}, I\_{\mathfrak{t}} + \theta\_{+})} \delta\_{I\_{\mathfrak{t}}} \left( \mathfrak{L}(\mathbb{X}\_{+}) \right) [\widetilde{p}\_{\mathbb{Z}\_{+}}^{(\overline{\mathfrak{s}}\_{\mathfrak{t}}\theta\_{+})}]^{\mathbb{X}\_{+}} \right. \tag{20}$$

$$
\boldsymbol{\omega}\_{Z\_{+}}^{(I\_{\boldsymbol{\upbeta}}\boldsymbol{I}\_{+},\boldsymbol{\upbeta}\_{+})} = [\boldsymbol{r}\_{\boldsymbol{B}+}]^{\mathbb{B}\_{+}\cap I\_{+}} [1 - \boldsymbol{r}\_{\boldsymbol{B}+}]^{\mathbb{B}\_{+}-I\_{+}} [\boldsymbol{p}\_{\boldsymbol{S}}]^{I\cap I\_{+}} [1 - \boldsymbol{p}\_{\boldsymbol{S}}]^{I-I\_{+}} [\boldsymbol{p}\_{\boldsymbol{T}}]^{\mathbb{B}\_{+}\cap I\_{+}} [1 - \boldsymbol{p}\_{\boldsymbol{T}}]^{\mathbb{B}\_{+}-I\_{+}} \boldsymbol{r}\_{\boldsymbol{S}}
$$

$$
\tilde{p}\_{Z\_+}^{(\xi,\theta\_+)}(\mathbf{x}\_+,l\_+) = \frac{\tilde{p}\_+^{(\xi)}(\mathbf{x}\_+,l\_+)\tilde{\Psi}\_{Z\_+}^{(\theta\_+,(l\_+))}(\mathbf{x}\_+,l\_+)}{\tilde{\Psi}\_{Z\_+}^{(\xi,\theta\_+)}(l\_+)}.
$$

$$
\tilde{p}\_{+}^{(\tilde{\xi})}(\mathbf{x}\_{+},l\_{+}) = \mathbf{1}\_{\mathbb{B}\_{+}}(\{l\_{+}\})p\_{\mathbb{B}\_{+}}(\mathbf{x}\_{+},l\_{+}) + \mathbf{1}\_{\mathbb{L}}(\{l\_{+}\})\tilde{p}\_{\mathbb{S}}^{(\tilde{\xi})}(\mathbf{x}\_{+},l\_{+}) + \mathbf{1}\_{\mathbb{S}}(\{l\_{+}\})\tilde{p}\_{T}^{(\tilde{\xi})}(\mathbf{x}\_{+},l\_{+}),
$$

$$
\vec{p}\_S^{(\xi)} = \frac{\langle p\_S(\cdot, l\_+) f\_{S+}(\mathbf{x}\_+|\cdot, l), p^{(\xi)}(\cdot, l\_+) \rangle}{\vec{p}\_S^{(\xi)}(l\_+)},
$$

$$
\vec{p}\_T^{(\xi)} = \frac{\langle p\_T(l\_+) f\_{T+}(\mathbf{x}\_+|\cdot, l), p^{(\xi)}(\cdot, l) \rangle}{\vec{p}\_T^{(\xi)}(l\_+)}
$$

$$p\_S^{(\xi)} = \langle p^{(\xi)}(\cdot, l), p\_S(l\_+) \rangle\_{\prime\prime}$$

$$
\bar{p}\_T^{(\xi)} = \langle p^{(\xi)}(\cdot, l), p\_T(l\_+) \rangle,
$$

$$
\Psi\_{Z\_+}^{(\xi, \theta\_+)}(l\_+) = \langle \bar{p}\_+^{(\xi)}(\cdot, l\_+), \Psi\_{Z\_+}^{(\theta\_+ \cdot (l\_+))}(\cdot, l\_+) \rangle.
$$

From this proposal density, Gibbs' sampler is applied to select high weight hypotheses. These hypotheses are subsequently used to form a standard GLMB density [30]:

$$\hat{\pi}(\mathbf{X}\_{+}|\mathbf{Z}\_{+}) = \Delta(\mathbf{X}\_{+}) \sum\_{l,\xi,l,\theta\_{+}} \delta\_{l+}(\mathcal{L}(\mathbf{X}\_{+})) \hat{\omega}\_{+}^{(I\_{z}^{\xi},l,\theta\_{+})}(\mathbf{Z}\_{+}) [p\_{B+} \boldsymbol{\upphi}\_{+}^{(\theta\_{+})}(\cdot|\mathbf{Z}\_{+})]^{\mathbf{Z}\_{+}} \frac{[\boldsymbol{\upphi}\_{+}^{(I\_{z}^{\xi},l,\theta\_{+})}(\cdot|\mathbf{Z}\_{+})]^{\mathbf{Z}\_{+},[\mathbf{Z}\_{+}]}}{[\boldsymbol{\upphi}\_{+}^{(I\_{z}^{\xi},l,\theta\_{+})}(\cdot|\mathbf{Z}\_{+})]^{\mathbf{I}\_{+}}} \tag{21}$$
 
$$\hat{\boldsymbol{\upomega}}\_{+}^{(I\_{z}^{\xi},l,\theta\_{+})}(\mathbf{Z}\_{+}) = \frac{\omega\_{+}^{(I\_{z}^{\xi})}(\boldsymbol{I}\_{+})[\bar{p}\_{+}^{(I\_{z}^{\xi},l,\theta\_{+})}(\cdot|\mathbf{Z}\_{+})]^{\mathbf{I}\_{+}}}{\sum\_{l,\xi,l,\theta\_{+}} \omega\_{+}^{(I\_{z}^{\xi})}(\boldsymbol{I}\_{+})[\bar{p}\_{+}^{(I\_{z}^{\xi},l,\theta\_{+})}(\cdot|\mathbf{Z}\_{+})]^{\mathbf{I}\_{+}}}$$

$$\mathfrak{gl}^{(I\_{\mathfrak{s}}^{\tau},l\_{+},\theta\_{+})}(\mathbf{x}\_{+},l\_{+}|\mathbf{Z}\_{+}) \triangleq \mathbf{1}\_{I\_{+}}(\{l\_{+}\}) \int p\_{+}^{(I\_{\mathfrak{s}}^{\tau}\theta,\mathfrak{s}\_{+})}(\{(\mathbf{x}\_{+},l\_{+}),(\mathbf{x}\_{1,+},l\_{1,+}),\ldots,(\mathbf{x}\_{n,+},l\_{n,+})\}|\mathbf{Z}\_{+}) d(\mathbf{x}\_{1,+},\ldots,\mathbf{x}\_{n,+})$$

$$\hat{p}^{(I\_{\mathcal{S}}^{\mathsf{r}},l\_{+},\theta\_{+})}(\mathbf{x}\_{+},l\_{+}|\mathbf{Z}\_{+}) \triangleq \mathbf{1}\_{\mathbb{B}\_{+}}(\{l\_{+}\})(p\_{\mathbb{B}+}(\cdot,l\_{+}),\boldsymbol{\upupmu}^{(\theta\_{+})}\_{+}(\cdot|\mathbf{Z}\_{+})) + (1-\mathbf{1}\_{\mathbb{B}\_{+}}(\{l\_{+}\}))\langle\hat{p}^{(I\_{\mathcal{S}}^{\mathsf{r}},l\_{+},\theta\_{+})}\_{\mathcal{Z}\_{+}}(\mathbf{x}\_{+},l\_{+}),1\rangle$$
