backward sampling algorithm [@fruhwirth1994data].

Note that the likelihood in (\[eq:interval likelihood\]) and the random walk prior process (\[eq:prior\]) define a state space model with non-linear and non-Gaussian observation model [@gordon1993novel], which allows to apply SMC inference methods. Generally, SMC methods are specifically designed for filtering problems [@doucet2000sequential] in state space models, where the main objective is to sample from $p(\boldsymbol{\beta}_{1:j}\text{|}\mathbf{t}_{1:j}),\,j=1,\ldots,J,$ sequentialy through lower-dimensional marginals, $p(\boldsymbol{\beta}_{j}\text{|\,}\mathbf{t}_{1:j})$, referred to as filtering distributions.

SMC methods for sampling from (\[eq:joint posterior\]) sequentially through the smoothing distribution, $p(\boldsymbol{\beta}_{j}\text{|}\mathbf{t}_{1:J})$, are also available in the literature. The forward-backward smoother samples parameters from the filtering distribution and smooths them in a backward procedure (see [@doucet2000sequential] and references therein). On the other hand, the two-filter smoother [@briers2010smoothing] and its computationally cheaper variant [@fearnhead2010sequential] combine samples from both a forward and a backward information filter. This paper builds on the algorithm of the two-filter smoothing suggested by @fearnhead2010sequential and the linear Bayes method [@west1985dynamic] to design an efficient inference scheme for PEH models.

The two-filter smoother\[subsec:The-two-filter-smoother\]
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The two-filter smothing recursion [@briers2010smoothing] reformulates the smoothing distribution as $$p\left(\boldsymbol{\beta}_{j}\text{|\,}\mathbf{t}_{1:J}\right)\propto p\left(\boldsymbol{\beta}_{j}\text{|\,}\mathbf{t}_{1:j-1}\right)p\left(\mathbf{t}_{j:J}|\,\boldsymbol{\beta}_{j}\right),\label{eq:Two-filter smoother}$$ where the first term on the right hand side, $$p\left(\boldsymbol{\beta}_{j}|\mathbf{t}_{1:j-1}\right)=\int p\left(\boldsymbol{\beta}_{j}|\boldsymbol{\beta}_{j-1}\right)p\left(\boldsymbol{\beta}_{j-1}|\mathbf{t}_{1:j-1}\right)d\boldsymbol{\beta}_{j-1},\label{eq:forward filter}$$ is the predictive prior recursion, and the second 