term, $$p\left(\mathbf{t}_{j:J}|\,\boldsymbol{\beta}_{j}\right)=L_{j}\left(\mathbf{t}_{j}|\boldsymbol{\beta}_{j}\right)\int p\left(\mathbf{t}_{(j+1):J}|\,\boldsymbol{\beta}_{j+1}\right)p\left(\boldsymbol{\beta}_{j+1}|\boldsymbol{\beta}_{j}\right)d\boldsymbol{\beta}_{j+1},\label{eq:Backward filter}$$ is the likelihood recursion. Both the expressions (\[eq:forward filter\]) and (\[eq:Backward filter\]) are computed recursively through two independent filters: i) the forward filter which estimates sequentially the filtering distribution $p(\boldsymbol{\beta}_{j-1}|\mathbf{t}_{1:j-1})$ forward in time, and ii) the backward filter which evaluates the likelihood recursion (\[eq:Backward filter\]) by mirroring the forward filter sequentially backward in time.

The forward filter begins from $j=1$ with a predifined initial distribution $p(\boldsymbol{\beta}_{1})$, and then proceeds propagating information forward in time through the usual posterior update: $$p\left(\boldsymbol{\beta}_{j}|\mathbf{t}_{1:j}\right)\propto L_{j}\left(\mathbf{t}_{j}|\boldsymbol{\beta}_{j}\right)p\left(\boldsymbol{\beta}_{j}|\mathbf{t}_{1:j-1}\right),\,j=2,\ldots,J.\label{eq:forward filter update}$$

On the other hand, the formulation of the backward filter is not straightforward. It turns out that (\[eq:Backward filter\]) is not a density for $\boldsymbol{\beta}_{j}$ and hence the integral may not be finite. To mirror the forward filter and ensure that the integral in (\[eq:Backward filter\]) is always finite, @briers2010smoothing introduce an artificial prior $\gamma_{j}\left(\boldsymbol{\beta}_{j}\right)$, for $j=J,\ldots,1$, so that the backward recursion expression becomes an artificial posterior; i.e, $$\begin{aligned}
\tilde{p}\left(\beta_{j}|\mathbf{t}_{j:J}\right) & \propto p\left(\mathbf{t}_{j:J}|\,\boldsymbol{\beta}_{j}\right)\gamma_{j}\left(\boldsymbol{\beta}_{j}\right)\label{eq: backward filtering distribution}\\
 & \propto L_{j}\left(\mathbf{t}_{j}|\boldsymbol{\beta}_{j}\right)\gamma_{j}\left(\boldsymbol{\beta}_{j}\right)\int\frac{\tilde{p}\left(\boldsymbol{\beta}_{j+1}|\mathbf{t}_{(j+1):J}\right)}{\gamma_{j+1}\left(\boldsymbol{\beta}_{j+1}\right)}p\left(\boldsymbol{\beta}_{j+1}|\boldsymbol{\beta}_{j}\right)d\boldsymbol{\beta}_{j+1},\nonumber \end{aligned}$$ where the first term in the inner expression of the integral is simply the likelihood $p(\mathbf{t}_{(j+1):J}|\,\boldsymbol{\beta}_{j+1})$. The backward filter starts at $J$ with the prior distribution $\gamma_{J}\left(\boldsymbol{\beta}_{J}\right)$ and proceeds by evaluating (\[eq: backward 