it remains to find a way of combining the two filters at each time point $j$. Instead of evaluating the double mixture (\[eq: particle smoothing distribution\]) which is computationally costly, @fearnhead2010sequential suggest running another particle filter that combines the samples from the forward and backward particle filters recursively. This approach relies on finding another proposal distribution $\bar{q}(\boldsymbol{\beta}_{j}|\boldsymbol{\beta}_{j-1}^{k},\mathbf{t}_{j},\tilde{\boldsymbol{\beta}}_{j+1}^{h})$ combining draws from the forward particle filter at the time point $j-1$ and the backward particle filter at $j+1$. New smoothing particles $\bar{\{\boldsymbol{\beta}_{j}}^{s}\}_{s=1}^{S}$are proposed from $\bar{q}$, and the corresponding smoothing importance weights $$\begin{aligned}
\bar{w}_{j}^{s} & \propto\frac{p\left(\bar{\boldsymbol{\beta}_{j}}^{s}|\boldsymbol{\beta}_{j-1}^{k}\right)L_{j}\left(\mathbf{t}_{j}|\bar{\boldsymbol{\beta}_{j}}^{s}\right)p\left(\tilde{\boldsymbol{\beta}}_{j+1}^{h}|\,\bar{\boldsymbol{\beta}_{j}}^{s}\right)w_{j-1}^{k}\tilde{w}_{j+1}^{h}}{\bar{q}\left(\bar{\boldsymbol{\beta}}_{j}|\boldsymbol{\beta}_{j-1}^{k},\mathbf{t}_{j},\tilde{\boldsymbol{\beta}}_{j+1}^{h}\right)\nu_{j}^{k}\,\tilde{\nu}_{j}^{h}\,\gamma_{j+1}\left(\tilde{\boldsymbol{\beta}}_{j+1}^{h}\right)},\,s=1,\ldots S\nonumber \\
 & \propto\frac{p\left(\bar{\boldsymbol{\beta}_{j}}^{s}|\boldsymbol{\beta}_{j-1}^{k}\right)L_{j}\left(\mathbf{t}_{j}|\bar{\boldsymbol{\beta}_{j}}^{s}\right)p\left(\tilde{\boldsymbol{\beta}}_{j+1}^{h}|\,\bar{\boldsymbol{\beta}_{j}}^{s}\right)}{L_{j}\left(\mathbf{t}_{j}\text{|\,\ensuremath{\boldsymbol{\beta}}}_{j-1}^{k}\right)\bar{q}\left(\bar{\boldsymbol{\beta}}_{j}|\boldsymbol{\beta}_{j-1}^{k},\mathbf{t}_{j},\tilde{\boldsymbol{\beta}}_{j+1}^{h}\right)L_{j}\left(\mathbf{t}_{j}|\,\tilde{\boldsymbol{\beta}}_{j+1}^{h}\right)\gamma_{j+1}\left(\tilde{\boldsymbol{\beta}}_{j+1}^{h}\right)},\label{eq: smoothing IW}\end{aligned}$$ approximate empirically the posterior distribution (\[eq: particle smoothing distribution\]). The two-filter smoother algorithm of @fearnhead2010sequential can be summarized in the following steps:

1\. Forward particle filter\
initialization: Sample $\beta_{0}^{k}\sim p\left(\beta_{0}\right)$ and set $w_{0}^{k}\propto \frac{1}{K}$\
2. Backward particle filter\
initialization: Sample $\beta_{J}^{h}\sim \gamma_{J}\left(\beta_{J}\right)$ and set $w_{J}^{k}\propto \frac{1}{K}$\
3. Smoothing step: Combining the two filters\
**Output:** $\{\bar{\boldsymbol{\beta}_{j}},\bar{w}_{j}\}_{j=1}^{J}$

Here, there are two things to be noted. First, the sample size $S$ of the smoothing particles need not be equal to the sample size for the forward (and backward) filtering particles $K$. One can gain computation time by setting $K$ small and $S$ bigger than 