Alternative forward sampling

If evaluation of $p(x_t\vert s_{t-1})$ or sampling from $p(s_t\vert s^i, x_t)$ is difficult or impossible alternative sampling methods can be obtained. Start again with the forward recursion (10) in the following form

$$p(s_t\vert x^t) \propto \sum_{s_{t-1}} p(x_t,s_t\vert s_{t-1}) \, p(s_{t-1}\vert x^{t-1})$$

$$= \sum_{s_{t-1}} \pi(s_t) \frac{p(x_t,s_t\vert s_t)}{\pi(s_t)} p(s_{t-1}\vert x^{t-1})$$

$$= \sum_i w^i \pi(s_t) \frac{p(x_t,s_t\vert s^i)}{\pi(s_t)} p(s^i\vert x^{t-1})$$