*2.4. Labeled Multi-Bernoulli RFS*

A labeled-random finite set (L-RFS) [34,35] means that each state of the RFS has a unique tag. This means we attach a unique label *<sup>l</sup>* <sup>∈</sup> <sup>L</sup> <sup>=</sup> {*α<sup>i</sup>* : *<sup>i</sup>* <sup>∈</sup> <sup>N</sup>} to each state *<sup>x</sup>* <sup>∈</sup> <sup>X</sup> where <sup>L</sup> is discrete countable space and N is the positive integer set space. The single target state is expressed as:

$$\mathbf{x}\_{k,N(k)} = \left(\mathbf{x}\_{k,N(k)}, l\_{k,N(k)}\right) \in \mathbb{X} \times \mathbb{L}.\tag{8}$$

The labels of the set <sup>X</sup> <sup>⊂</sup> <sup>X</sup> <sup>×</sup> <sup>L</sup> can be represented by <sup>L</sup> (X) <sup>=</sup> {L (x) : x <sup>∈</sup> <sup>X</sup>}, where <sup>L</sup> : <sup>X</sup> <sup>×</sup> <sup>L</sup> <sup>→</sup> <sup>L</sup> is defined by <sup>L</sup> ((*x*, *<sup>l</sup>*)) <sup>=</sup> *<sup>l</sup>*. The distinct label indicator is defined by <sup>Δ</sup> (X) <sup>=</sup> *<sup>δ</sup>*|X<sup>|</sup> (|L (X)|).

The parameter of a labeled multi-Bernoulli(LMB) RFS can be described as a set {(*r*(*ζ*), *<sup>p</sup>*(*ζ*)) : *<sup>ζ</sup>* <sup>∈</sup> <sup>Ψ</sup>} with index set <sup>Ψ</sup>. We extend the problem on space<sup>X</sup> to space<sup>X</sup> <sup>×</sup> <sup>L</sup>, thus, the probability density distribution of labeled LMB-RFS is given by [34]:

$$\pi\left(\left\{\left(\mathbf{x}\_{1},l\_{1}\right),\ldots,\left(\mathbf{x}\_{n},l\_{n}\right)\right\}\right) = \delta\_{\mathbb{R}}\left(\left\lfloor\left\{l\_{1},\ldots,l\_{n}\right\}\right\rfloor\right)\prod\_{\boldsymbol{\zeta}\in\mathbf{Y}}\left(1-r^{\left(\boldsymbol{\zeta}\right)}\right)\prod\_{j=1}^{n}\frac{\mathbf{1}\_{\mathbf{z}\left(\mathbf{y}\right)}\left(l\_{j}\right)r^{\left(a^{-1}\left(l\_{j}\right)\right)}p^{\left(a^{-1}\left(l\_{j}\right)\right)}\left(\mathbf{x}\_{j}\right)}{1-r^{\left(a^{-1}\left(l\_{j}\right)\cdot 1\right)}}\tag{9}$$

The following simplified alternative form of the LMB can be simplified as:

$$
\pi\left(X\right) = \Delta\left(X\right)\mathbf{1}\_{a\left(\Psi\right)}\left(\mathcal{L}\left(X\right)\right)\left[p\left(\cdot\right)\right]^{X} \tag{10}
$$
