filtering distribution\]) recursively backward in time, for $j=J-1,\ldots,1$ .

The smoothing distribution $p(\boldsymbol{\beta}_{j}|\mathbf{t}_{1:J})$ is therefore estimated by combining the forward filter standing at $j-1$ and the backward filter standing at $j+1$ (see Algorithm $1$). If the likelihood is Gaussian and linear then both filters are analytically tractable. Otherwise, some form of approximation such as the particle filter approach [@doucet2000sequential; @arulampalam2002tutorial] is needed.

### Particle filtering approximation

Particle filters provide a recursive procedure of approximating the forward filtering distribution (\[eq:forward filter update\]) by an empirical distribution defined on a finite sample of points $\{\boldsymbol{\beta}_{j}^{k}\}_{k=1}^{K}$ (commonly known as particles) weighted by the probability masses $\{w_{j}^{k}\}_{k=1}^{K}$ (importance weights). Assuming that a sample of particles and their corresponding importance weights at $j-1$ are available, then the predictive prior (\[eq:forward filter\]) is approximated as $$p\left(\boldsymbol{\beta}_{j}|t_{1:j-1}\right)\propto\sum_{k=1}^{K}p\left(\boldsymbol{\beta}_{j}|\boldsymbol{\beta}_{j-1}^{k}\right)w_{j-1}^{k}.\label{eq:predictive prior approx}$$

Similarly, the backward filtering distribution (\[eq: backward filtering distribution\]) is approximated empirically by a finite sample of particles $\{\tilde{\boldsymbol{\beta}}_{j}^{k}\}_{k=1}^{K}$ weighted by $\{\tilde{w}_{j}^{k}\}_{k=1}^{K}$, and if the backward filter stands at the time point $j+1$, then the likelihood (\[eq:Backward filter\]) is approximated as $$p\left(\mathbf{t}_{j:J}|\,\boldsymbol{\beta}_{j}\right)\propto L_{j}\left(\mathbf{t}_{j}|\boldsymbol{\beta}_{j}\right)\sum_{h=1}^{K}\frac{p\left(\tilde{\boldsymbol{\beta}}_{j+1}^{h}|\boldsymbol{\beta}_{j}\right)}{\gamma_{j+1}\left(\tilde{\boldsymbol{\beta}}_{j+1}^{h}\right)}\tilde{w}_{j+1}^{h}.\label{eq:Likelihood recurrsion approx}$$

Therefore, realizations from the forward particle filter standing at the time point $j-1$ and the backward filter at $j+1$ can be combined to 