Algorithm 3.

Backward Simulation Particle Smoother

1:	/* simulate T smoothed trajectories from time N */
2:	for k = 1 to T do
3:	/* draw index ℓ according to weights ${\left\{w_N^j\right\}}_{j=1}^M$ */
4:	set ${\tilde{x}}_{N\mid N}^k={\tilde{x}}_N^{\ell }$.
5:	end for
6:	/* propagate the remaining simulated trajectories */
7:	for t = N − 1 to 1 do
8:	for k = 1 : T do
9:	/* compute new weights */
10:	${\tilde{w}}_{t\mid N}^{j,k}\propto w_t^{j}f\left({\tilde{x}}_{t+1\mid N}^k\mid x_t^j\right)$ for j = 1, …, M.
11:	/* normalize the smoothing weights */
12:	${\tilde{w}}_{t\mid N}^{j,k}=\frac{{\tilde{w}}_{t\mid N}^{j,k}}{{\sum }_{j=1}^M{\tilde{w}}_{t\mid N}^{j,k}},j=1,\dots ,M$.
13:	/* draw index ℓ according to ${\left\{{\tilde{w}}_{t\mid N}^{j,k}\right\}}_{j=1}^M$ */
14:	set ${\tilde{x}}_{t\mid N}^k={\tilde{x}}_t^{\ell }$.
15:	end for
16:	end for