*2.3. Multi-Bernoulli RFS*

The state of the target and the measurements are random variables. The Bernoulli distribution can be used to describe a single target *<sup>X</sup>* <sup>∈</sup> <sup>X</sup>. Hence, a singleton target probability *<sup>X</sup>* is *<sup>r</sup>* which satisfies the spatial distribution of a probability density *p* (*x*) and the probability that the target does not exist is 1 − *r*. The probability density distribution of the Bernoulli RFS is written as follows:

$$\pi \left( \mathbf{x} \right) = \begin{cases} 1 - r & \mathbf{X} = \bigcirc \\ r \cdot p \left( \mathbf{x} \right) & \mathbf{X} = \{ \mathbf{x} \} \\ 0 & \text{otherwise} \end{cases} \tag{6}$$

The Multi-Bernoulli RFS is given by combining of the *<sup>M</sup>* independent Bernoulli RFS *<sup>X</sup>*(*i*) <sup>∈</sup> <sup>X</sup>, *<sup>i</sup>* <sup>=</sup> 1, ··· , *<sup>M</sup>*(satisfying *<sup>X</sup>* <sup>=</sup> & *M i*=1 *<sup>X</sup>*(*i*)) with existence probability *<sup>r</sup>*(*i*) <sup>∈</sup> (0, 1), which is described by *r*(*i*), *<sup>p</sup>*(*i*) *<sup>M</sup> i*=1 . ∑*<sup>M</sup> <sup>i</sup>* = 1 *<sup>r</sup>*(*i*) is the mean cardinality of the Multi-Bernoulli RFS. Therefore, the probability density distribution of multi-Bernoulli is expressed by [12,18]:

$$\pi\left(\left\{\mathbf{x}\_1,\dots,\mathbf{x}\_n\right\}\right) = \prod\_{j=1}^{M} \left(1 - r^{(j)}\right) \sum\_{1 \le i\_1 \ne \dots \ne i\_n \le M} \prod\_{j=1}^{n} \frac{r^{\binom{i\_j}{j}} p^{\binom{i\_j}{j}} \left(\mathbf{x}\_j\right)}{1 - r^{\binom{i\_j}{j}}}.\tag{7}$$
