2.2.1. Prediction Step

If the posterior probability density at time *<sup>k</sup>* <sup>−</sup> 1 is <sup>π</sup>*k*−1(*X*) = {(*<sup>r</sup>* (*i*) *k*−1 , *p* (*i*) *<sup>k</sup>*−1(*xk*−1))} *Mk*−<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> , then the predicted multi-target density is also a multi-Bernoulli formed by the union of the multi-Bernoulli for the surviving targets and target births

$$\pi\_{k|k-1} = \left\{ (r\_{P,k|k-1}^{(i)}, p\_{P,k|k-1}^{(i)}(\mathbf{x}\_k)) \right\}\_{i=1}^{M\_{k-1}} \cup \left\{ (r\_{\Gamma,k|k-1}^{(i)}, p\_{\Gamma,k|k-1}^{(i)}(\mathbf{x}\_k)) \right\}\_{i=1}^{M\_{\Gamma,k}} \tag{13}$$

where (*r* (*i*) Γ,*k*|*k*−1 , *p* (*i*) <sup>Γ</sup>,*k*|*k*−1(*xk*)) *<sup>M</sup>*Γ,*<sup>k</sup> i*=1 is the predicted multi-Bernoulli for the target births and it is usually assumed to be known. (*r* (*i*) *P*,*k*|*k*−1 , *p* (*i*) *<sup>P</sup>*,*k*|*k*−1(*xk*)) *Mk*−<sup>1</sup> *i*=1 is the predicted multi-Bernoulli for the surviving targets and it is given by

$$r\_{P,k\mid k-1}^{(i)} = r\_{k-1}^{(i)} \left\langle p\_{k-1}^{(i)}(\mathbf{x}\_{k-1}), p\_{S,k}(\mathbf{x}\_{k-1}) \right\rangle \tag{14}$$

$$p\_{P,k|k-1}^{(i)}(\mathbf{x}\_k) = \frac{\left\langle f\_{k|k-1}(\mathbf{x}\_k|\mathbf{x}\_{k-1}), p\_{k-1}^{(i)}(\mathbf{x}\_{k-1})p\_{S,k}(\mathbf{x}\_{k-1}) \right\rangle}{\left\langle p\_{k-1}^{(i)}(\mathbf{x}\_{k-1}), p\_{S,k}(\mathbf{x}\_{k-1}) \right\rangle} \tag{15}$$

where *f*, *g* <sup>=</sup> # *<sup>f</sup>*(*x*)*g*(*x*)*dx* denotes the inner product operation, *pS*,*k*(*xk*−1) is the survival probability of the surviving targets, *fk*|*k*−1(*xk*|*xk*−1) is the single-target transition density. There are *Mk*|*k*−<sup>1</sup> <sup>=</sup> *Mk*−<sup>1</sup> + *M*Γ,*k*−<sup>1</sup> predicted hypothesized tracks.
