particle filter which propagates information backwards in time, and a particle filter which combines the former two particle filters. These algorithms require a proposal distribution (importance distribution) that is easy to sample from. The so called bootstrap particle filters propose particles (parameter draws) from the prior distribution. This approach is prone to high degeneracy of the particles because the proposal does not incorporate any evidence from data. The standard way of incorportating evidence from data in the proposal distribution relies on the second order Taylor series expension of the log likelihood [@fearnhead2010sequential]. However, this is not appropriate for the PEH models because the mode and the hessian of the likelihood are not finite for censored individuals.

The main contribution of this paper is an efficient class of proposal distributions specially adapted to the nature of survival data and PEH models. The proposal of the forward filter is designed using the Laplace approximation of the posterior distribution of the hazard function with respect to the linear predictor (the log of the hazard function) and the linear Bayes method of @west1985dynamic. The proposal distributions of the other filters follow the linear Bayes theory. Thus, the proposed inference algorithm is referred to as 