*2.4. Multitarget Bayesian Recursion*

Let *πk*−<sup>1</sup> (·|*Z*1:*k*−1) denotes the multitarget density of the multitarget state at time *k* − 1, where *<sup>Z</sup>*1:*k*−<sup>1</sup> = (*Z*1, ... , *Zk*−1) is the set of all observation history up to time *<sup>k</sup>* − 1. For simplicity, we obmit the dependence on past measurements, i.e, we use *πk*−<sup>1</sup> (·|*Zk*−1) instead of *πk*−<sup>1</sup> (·|*Z*1:*k*−1). The multitarget Bayes filter use the Chapman-Kolmogorov equation to predict the multitarget state to time *k* given posterior at time *k* − 1 as follows [3]

$$
\pi\_{k|k-1} \left( \mathbf{X}\_k | Z\_{k-1} \right) = \int f\_{k|k-1} \left( \mathbf{X}\_k | \mathbf{X} \right) \pi\_{k-1} \left( \mathbf{X} | Z\_{k-1} \right) d\mathbf{X}, \tag{18}
$$

where *fk*|*k*−<sup>1</sup> (*Xk*|*X*) is defined as the multitarget transition kernel from time *<sup>k</sup>* <sup>−</sup> 1 to time *<sup>k</sup>*, and the integral in Equation (18) is the set integral defined for any function *<sup>f</sup>* : <sup>F</sup> (<sup>X</sup> <sup>×</sup> <sup>L</sup>) <sup>→</sup> <sup>R</sup>,

$$\int f\left(\mathbf{X}\right)\delta\mathbf{X} = \sum\_{i=0}^{\infty} \frac{1}{i!} \int f\left(\{\mathbf{x}\_1, \dots, \mathbf{x}\_i\}\right) d\left(\mathbf{x}\_1, \dots, \mathbf{x}\_i\right). \tag{19}$$

The multitarget state *X<sup>k</sup>* is partially observed at time *k*, and the RFS *Zk* is modeled by the multitarget likelihood function *gk* (*Zk*|*Xk*), thus the multitarget posterior at this time is given by Bayes rule:

$$
\pi\_k \left( \mathbf{X}\_k | \mathbf{Z}\_k \right) = \frac{g\_k \left( \mathbf{Z}\_k | \mathbf{X}\_k \right) \pi\_{k|k-1} \left( \mathbf{X}\_k | \mathbf{Z}\_{k-1} \right)}{\int g\_k \left( \mathbf{Z}\_k | \mathbf{X} \right) \pi\_{k|k-1} \left( \mathbf{X} | \mathbf{Z}\_{k-1} \right) d\mathbf{X}}. \tag{20}
$$
