Efficient particle smoothing for Bayesian inference in dynamic survival models

Abstract

This article proposes an efficient Bayesian inference for piecewise exponential hazard (PEH) models, which allow the effect of a covariate on the survival time to vary over time. The proposed inference methodology is based on a particle smoothing algorithm that depends on three particle filters. Efficient proposal (importance) distributions for the particle filters tailored to the nature of survival data and PEH models are developed using the Laplace approximation of the posterior distribution and linear Bayes theory. The algorithm is applied to both simulated and real data, and the results show that it is faster and more efficient than a state-of-the-art MCMC sampler, and scales well in high-dimensional and relatively large data.

Details on the approximation of the posterior moments of the linear predictor

Given the structure of the likelihood (6), the model parameter $\uplambda _{j}$ has a conjugate $\text {Gamma}(\alpha _{j},\psi _{j})$ prior distribution, which implies that the marginal posterior of $\uplambda _{j}$ is $\text {Gamma}(\alpha _{j}+d_{j},\psi _{j}+t_{j})$. Taking into account the Jacobian of the transformation $\eta _{j}=\ln \uplambda _{j}$, it can be shown that

$$\begin{aligned} p\left( \eta _{j}|\varvec{\beta }_{j-1},{\mathbf {t}}_{1:j}\right) \propto \exp \left\{ \eta _{j}\left( \alpha _{j}+d_{j}\right) -\left( \psi _{j}+t_{j}\right) \exp \left\{ \eta _{j}\right\} \right\} , \end{aligned}$$

(27)

where ${\mathbf {t}}_{1:i,j}$ is the set of exposure times for the first i individuals observed in the interval $I_{j}$. In order to apply the conditional expectations (21), we need to compute the first and second derivatives of the log of (27)

$$\begin{aligned} \dfrac{\partial \ln p\left( \eta _{j}|\varvec{\beta }_{j-1},{\mathbf {t}}_{1:j}\right) }{\partial \eta _{j}}=\alpha _{j}+d_{j}-\left( \psi _{j}+t_{j}\right) \exp \left\{ \eta _{j}\right\} , \end{aligned}$$

and

$$\begin{aligned} \dfrac{\partial ^{2}\ln p\left( \eta _{j}|\varvec{\beta }_{j-1},{\mathbf {t}}_{1:j}\right) }{\partial \eta _{j}^{2}}=-\left( \psi _{j}+t_{j}\right) \exp \left\{ \eta _{j}\right\} . \end{aligned}$$

From the first derivative, one can show that the mode lies at $\hat{\eta _{j}}=\ln (\frac{\alpha _{j}+d_{j}}{\psi _{j}+t_{j}})$, which lead to the final expressions

$$\begin{aligned} E[\eta _{j}|\varvec{\beta }_{j-1},{\mathbf {t}}_{j}]&=\left. \dfrac{\partial \ln p\left( \eta _{j}|\varvec{\beta }_{j-1},{\mathbf {t}}_{1:j}\right) }{\partial \eta _{j}}\right| _{\eta _{j}=\hat{\eta _{j}}}=\ln \left( \frac{\alpha _{j}+d_{j}}{\psi _{j}+t_{j}}\right) ,\\ V[\eta _{j}|\varvec{\beta }_{j-1},{\mathbf {t}}_{j}]&=\left. \dfrac{\partial ^{2}\ln p\left( \eta _{j}|\varvec{\beta }_{j-1},{\mathbf {t}}_{1:j}\right) }{\partial \eta _{j}^{2}}\right| _{\eta _{j}=\hat{\eta _{j}}}^{-1}=\frac{1}{\alpha _{j}+d_{j}} \end{aligned}$$

The hyper-parameters $\alpha _{j}$ and $\psi _{j}$ are selected in order to match the true moments of the prior with the moments from the deterministic relationship $\eta _{j}={\mathbf {z}}^{\prime }\varvec{\beta }_{j-1}$. This is accomplished by setting $\ln \alpha _{j}-\ln \psi _{j}={\mathbf {z}}^{\prime }\varvec{\beta }_{j-1}$ and $\alpha _{j}^{-1}=Q_{j}$; hence $\psi _{j}=Q_{j}^{-1}\exp \{-{\mathbf {z}}_{i}^{\prime }\varvec{\beta }_{j-1}\}$, which proves the moments of the proposal q described in Sect. 3.3.