*4.3. The Multi-Sensor Likelihood*

Given the multi-target state *X*, each (*x*, *l*) ∈ *X* is either detected with probability *pD*,*<sup>m</sup>* (*x*, *l*) and generates observation *z* with likelihood function *g* (*z* |*x*, *l*). For *S* sensors, the multi-sensor and multi-target mapping [56] is defined by *<sup>θ</sup>*(*m*) : <sup>L</sup> <sup>→</sup> 0, 1 ··· , *Z*(*m*) , *m* = 1, ..., *S*. The set Θ represents the space of vector maps *θ* = (*θ*(1), ..., *θ*(*S*)). Assuming that the target and clutter generation are independent and the multi-sensor likelihood function is given by [34]:

$$\log\left(Z|X\right) \propto \sum\_{\theta \in \Theta(\mathcal{L}(X))} \left[\psi\_{Z\_1}\left(\cdot;\theta^{(1)}\right)\right]^X \cdot \dots \left[\psi\_{Z\_m}\left(\cdot;\theta^{(S)}\right)\right]^X$$

$$\Psi\_{Z\_m}(\mathbf{x},l;\theta) = \begin{cases} \frac{p\_{D,m}(\mathbf{x},l)\mathfrak{z}\left(z\_{\theta(l,m)}|\mathbf{x},l\right)}{\mathcal{K}\_m\left(z\_{\theta(l,m)}\right)}, & \text{if } \theta\left(l,m\right) > 0\\ 1 - p\_{D,m}\left(\mathbf{x},l\right), & \text{if } \theta\left(l,m\right) = 0 \end{cases},$$

where *pD*,*<sup>m</sup>* (*x*, *l*) is the probability detection, K*<sup>m</sup>* is Poisson clutter, for sensor *m*.
