*2.4. The Single Object RTS Smoother*

Given a set single object observation {*z*1:*N*}, where *N* ≤ *K* with *K* is the total number of tracking time steps, the smoothed density of an object state at time *k* ≤ *N*, *p*(*xk*|*z*1:*N*), is obtained as follows [36].

Initially, let the joint distribution of *xk* and *xk*<sup>+</sup><sup>1</sup> be rewritten as follows:

$$p(\mathbf{x}\_{k'}\mathbf{x}\_{k+1}|\mathbf{z}\_{1:k}) = p(\mathbf{x}\_{k+1}|\mathbf{x}\_k)p(\mathbf{x}\_k|\mathbf{z}\_{1:k}) \tag{12}$$

Then, the distribution of *xk* given *xk*<sup>+</sup><sup>1</sup> and *z*1:*<sup>k</sup>* is given as follows:

$$p(\mathbf{x}\_k | \mathbf{x}\_{k+1}, z\_{1:k}) = \frac{p(\mathbf{x}\_k, \mathbf{x}\_{k+1} | z\_{1:k})}{p(\mathbf{x}\_{k+1} | z\_{1:k})} \tag{13}$$

where *<sup>p</sup>*(*xk*<sup>+</sup>1|*z*1:*k*) = \$ *<sup>p</sup>*(*xk*<sup>+</sup>1|*xk*)*p*(*xk*|*z*1:*k*)*dxk*

From the Markov state-space model, we have the following property: *p*(*xk*|*xk*+1, *z*1:*N*) = *p*(*xk*|*xk*+1, *z*1:*k*). Hence, we have the following:

$$p(\mathbf{x}\_k | \mathbf{x}\_{k+1}, z\_{1:N}) = \frac{p(\mathbf{x}\_{k'} \mathbf{x}\_{k+1} | z\_{1:k})}{p(\mathbf{x}\_{k+1} | z\_{1:k})} \tag{14}$$

Then, the joint distribution of *xk* and *xk*<sup>+</sup><sup>1</sup> given the measurements set *z*1:*<sup>N</sup>* is given as follows:

$$p(\mathbf{x}\_{k}, \mathbf{x}\_{k+1}|z\_{1:N}) = p(\mathbf{x}\_{k}|\mathbf{x}\_{k+1}, z\_{1:N}) p(\mathbf{x}\_{k+1}|z\_{1:N}) \tag{15}$$

Finally, the smoothed density of state *xk* can then be obtained via the marginalization step as follows:

$$p(\mathbf{x}\_k|z\_{1:N}) = \int p(\mathbf{x}\_k|\mathbf{x}\_{k+1}, z\_{1:N}) p(\mathbf{x}\_{k+1}|z\_{1:N}) d\mathbf{x}\_{k+1} \tag{16}$$
