$K$. More specifically by setting $S=RK$ ( in the following section $R=2$ in all runs of PSLiB). Second, the smoothing weights depend on the artificial prior $\gamma_{j+1}$, which can be any distribution. In order to ensure that the particles from backward filter are sampled from the smoothing distribution, @fearnhead2010sequential suggest setting $\gamma_{j+1}(\boldsymbol{\beta}_{j+1})=p(\boldsymbol{\beta}_{j+1}|\mathbf{t}_{1:j})$.

Given the particles sampled at the time point $j$ and their corresponding importance weights, this prior can be represented by the mixture in (\[eq:predictive prior approx\]); however, doing this induces extra computational costs. To avoid it, one can use the linear Bayes method of @west1985dynamic to approximate the mixture by a single Gaussian distribution (see Appendix \[sec:The-artificial-prior\]).

### Proposal distribution based on linear Bayes method\[subsec:Proposal-distribution\]

The aim of this section is to delineate the proposal distributions $q$, $\tilde{q}$ and $\bar{q}$. Since the regression coefficients are continuous random variables, it is convinient to construct $q$ as a mixture of Gaussian component distributions $$q(\boldsymbol{\beta}_{j}|\boldsymbol{\beta}_{j-1},\mathbf{t}_{j})\simeq N_{P+1}(\boldsymbol{\beta}_{j}|\mathbf{m}_{j},\mathbf{C}_{j}),\label{eq:Forward proposal approximation}$$ where $\mathbf{m}_{j}$ and $\mathbf{C}_{j}$ are respectively the mean and covariance matrix of the component distributions of the mixture (\[eq: APF proposal\]), and $P$ the dimension of the covariate vector. The standard APF proposes particles from the random walk process (\[eq:prior\]), which means that 