approximate the smoothing distribution (\[eq:Two-filter smoother\]) as $$p\left(\boldsymbol{\beta}_{j}\text{|\,}\mathbf{t}_{1:J}\right)\propto\sum_{k=1}^{K}\sum_{h=1}^{K}p\left(\boldsymbol{\beta}_{j}|\boldsymbol{\beta}_{j-1}^{k}\right)L_{j}\left(\mathbf{t}_{j}|\boldsymbol{\beta}_{j}\right)\frac{p\left(\tilde{\boldsymbol{\beta}}_{j+1}^{h}|\boldsymbol{\beta}_{j}\right)}{\gamma_{j+1}\left(\tilde{\boldsymbol{\beta}}_{j+1}^{h}\right)}w_{j-1}^{k}\tilde{w}_{j+1}^{h}.\label{eq:  particle smoothing distribution}$$

The posterior (\[eq: particle smoothing distribution\]) requires estimates of the forward and backward importance weights. Using importance sampling [@gordon1993novel], the importance weights are computed recursively as the ratio of the filtering distribution and the proposal distribution $q$ (for the forward filter) and $\tilde{q}$ (for the backward filter). Considering the auxiliary particle filter (APF) of @pitt1999filtering, particles are proposed from the mixture distribution, $$q\left(\boldsymbol{\beta}_{j}|\mathbf{t}_{1:j}\right)=\sum_{k=1}^{K}\nu_{j}^{k}q\left(\boldsymbol{\beta}_{j}|\boldsymbol{\beta}_{j-1}^{k},\mathbf{t}_{j}\right),\label{eq: APF proposal}$$ for the forward filter, and $$\tilde{q}\left(\tilde{\boldsymbol{\beta}}_{j}|\mathbf{t}_{j:J}\right)=\sum_{h=1}^{K}\tilde{\nu}_{j}^{h}\tilde{q}\left(\tilde{\boldsymbol{\beta}}_{j}|\tilde{\boldsymbol{\beta}}_{j+1}^{h},\mathbf{t}_{j}\right),\label{eq:APF backward proposal}$$ for the backward filter. Where $\left\{ \nu_{j}^{h}\right\} _{h=1}^{K}$ and $\left\{ \tilde{\nu}_{j}^{k}\right\} _{h=1}^{K}$ are some normalized mixture weights. To sample from these proposal distributions a mixture component $a_{k}$ (referred to as ancestor) of $q$ is selected with probability proportional to $\nu_{j}$ and then a particle $\boldsymbol{\beta}_{j}^{k}$ is proposed from $q(\boldsymbol{\beta}_{j}|\boldsymbol{\beta}_{j-1}^{a_{k}},\mathbf{t}_{j})$; a similar procedure is used for $\tilde{q}$. Therefore, setting $\nu_{j}^{k}\wasypropto L_{j}(\mathbf{t}_{j}\text{|\ensuremath{\boldsymbol{\beta}}}_{j-1}^{k})w_{j-1}^{k}$ (and similarly $\tilde{\nu}_{j}^{h}\wasypropto L_{j}(\mathbf{t}_{j}\text{|\ensuremath{\beta}}_{j+1}^{h})\tilde{w}_{j+1}^{h}$ for the backward filter) implies that one proposes only from ancestors that have high importance weights and high predictive density.

Given the ancestors $a_{k}$ and $\tilde{a}_{h}$ from the forward and backward filters respectively, the corresponding importance weights become, $$\begin{aligned}
w_{j}^{k} & \wasypropto\frac{p\left(\boldsymbol{\beta}_{j}^{k}|\,\mathbf{t}_{1:j}\right)}{q\left(\boldsymbol{\beta}_{j}^{k}|\,\mathbf{t}_{1:j}\right)}=\frac{L_{j}\left(\mathbf{t}_{j}|\,\boldsymbol{\beta}_{j}^{k}\right)p\left(\boldsymbol{\beta}_{j}^{k}|\,\boldsymbol{\beta}_{j-1}^{a_{k}}\right)}{L_{j}\left(\mathbf{t}_{j}\text{|\,\ensuremath{\boldsymbol{\beta}}}_{j-1}^{k}\right)q\left(\boldsymbol{\beta}_{j}^{k}|\,\boldsymbol{\beta}_{j-1}^{a_{k}},\mathbf{t}_{j}\right)},\nonumber \\
\tilde{w}_{j}^{h} & \wasypropto\frac{\tilde{p}\left(\tilde{\boldsymbol{\beta}}_{j}^{h}|\,\mathbf{t}_{j:J}\right)}{\tilde{q}\left(\tilde{\boldsymbol{\beta}}_{j}^{h}|\,\mathbf{t}_{j:J}\right)}=\frac{L_{j}\left(\mathbf{t}_{j}|\,\tilde{\boldsymbol{\beta}}_{j}^{h}\right)p\left(\tilde{\boldsymbol{\beta}}_{j+1}^{\tilde{a}_{h}}|\,\tilde{\boldsymbol{\beta}}_{j}^{h}\right)\gamma_{j}\left(\tilde{\boldsymbol{\beta}}_{j}^{h}\right)}{L_{j}\left(\mathbf{t}_{j}|\,\tilde{\boldsymbol{\beta}}_{j+1}^{\tilde{a}_{h}}\right)\widetilde{q}\left(\tilde{\boldsymbol{\beta}}_{j}^{h}|\,\tilde{\boldsymbol{\beta}}_{j+1}^{\tilde{a}_{h}},\mathbf{t}_{j}\right)\gamma_{j+1}\left(\tilde{\boldsymbol{\beta}}_{j+1}^{\tilde{a}_{h}}\right)}.\label{eq: IW forward and backward PF}\end{aligned}$$

Finally, 