$\mathbf{m}_{j}=\boldsymbol{\beta}_{j-1}$ and $\mathbf{C}_{j}=\mathbf{U}_{j}$. This choice is prone to high degeneracy since particles are proposed from the prior. In addition to that, the discount factor approach, which defines $\mathbf{U}_{j}=(\phi^{-1}-1)\boldsymbol{\Sigma}_{j-1}$, makes this proposal impractical as it will be hard to control the variance of the proposal. To obtain a better proposal distribution it is necessary to include evidence from the data; one way to do this, is to linearize the likelihood locally through a second order Tylor series expansion of the log likelihood [@fearnhead2010sequential] w.r.t the linear predictor $\eta_{j}$ around the mode value of $\eta_{j}$. For the PEH model, the mode of the linear predictor for an individual $i$ is $\hat{\eta}_{ij}=\log(\frac{d_{ij}}{t_{ij}})$ and the hessian is $d_{ij}^{-1}$. Hence, this linearization is not guaranteed to work since the mode and the hessian are not always finite – both $d_{ij}$ and $t_{ij}$ can be zero.

One alternative to get around this issue would be to first update the parameter $\lambda_{j}$ (the index $i$ is omitted for notational simplicity) and then exploit the fact that $\boldsymbol{\beta}_{j}$ enters the likelihood through the linear predictor, $\boldsymbol{\eta}_{j}=\ln\lambda_{j}=\mathbf{z}^{\prime}\mathbf{\boldsymbol{\beta}}_{j}$. Since $\eta_{j}$ and $\boldsymbol{\beta}_{j}$ are connected deterministically, one needs the posterior estimates $E[\eta_{j}|\mathbf{t}_{j}]$ and $V[\eta_{j}|\mathbf{t}_{j}]$ to update the mean and variance of 