Given a state-space model
\begin{align} x_k &= f_k(x_{k-1}, v_{k-1}),\\ z_k &= h_k(x_k, w_k), \end{align}
where $x_k \in {\mathbb R}^{n}$ and $y_k \in {\mathbb R}^{m}$ are the system state and system output in time $k \in \overline{\mathbb Z}_+$. $v_{k-1} \in {\mathbb R}^{p}$ and $w_k \in {\mathbb R}^{q}$ are white noises. $f_k:{\mathbb R}^{n+p} \to {\mathbb R}^{n}$ and $h_k:{\mathbb R}^{n+q} \to {\mathbb R}^{m}$ are nonlinear functions.
Then, the ISI method can be used to approximate the posterior $p(x_k|z_k,\ldots,z_1)$. However, it needs the transition probability $p(x_k|x_{k-1})$. In many textbooks, this probability is assumed to be given, but how to derive it from the above two highly nonlinear equations in the state-space model? Thanks very much!