for these elements is based on the Watanabe-Akaike information creterion (WAIC); see @gelman2014understanding. The WAIC uses the posterior sample and an out-of-sample test set to estimate the log posterior predictive density with an adjustment for the effective number of parameters. Given an out-of-sample test set of size $n_{\textrm{test}}$ $$\begin{aligned}
WAIC & =\sum_{i=1}^{n_{\textrm{test}}}\log E_{\textrm{post}}\left[L\left(t_{i}^{*}|\boldsymbol{\beta}_{1:J}\right)\right]-V_{\textrm{post}}\left[\log L\left(t_{i}^{*}|\boldsymbol{\beta}_{1:J}\right)\right]\label{eq:WAIC}\\
 & =\sum_{i=1}^{n_{\textrm{test}}}\left\{ \log\left(E_{\textrm{post}}\left[L\left(t_{i}^{*}|\boldsymbol{\beta}_{1:J}\right)\right]\right)-E_{\textrm{post}}\left[\left(\log L\left(t_{i}^{*}|\boldsymbol{\beta}_{1:J}\right)-E_{\textrm{post}}\left[\log L\left(t_{i}^{*}|\boldsymbol{\beta}_{1:J}\right)\right]\right)^{2}\right]\right\} .\nonumber \end{aligned}$$

Given a sample of particles $\{\boldsymbol{\beta}_{1:J}^{k}\}_{k=1}^{K}$ from the smoothing posterior distribution and their corresponding smoothing importance weights, $\{\bar{w}_{J}^{k}\}_{k=1}^{K}$ (note that $\bar{w}_{J}$ are the weights for the complete paths $\boldsymbol{\beta}_{1:J}$), the expectations in (\[eq:WAIC\]) can be approximated as $$E_{\textrm{post}}\left[g\left(t_{i}|\beta_{1:J}\right)\right]=\frac{\sum_{k=1}^{K}g\left(t_{i}|\beta_{1:J}^{k}\right)\bar{w}_{J}^{k}}{\sum_{k=1}^{K}\bar{w}_{J}^{k}},\label{eq:IS approximation of expected values}$$ where $g$ is any transformation of the likelihood function. Further, the prediction of the probability that an individual $i$ survives up to the time $t$ is approximated by the particle smoother as $$\begin{aligned}
\hat{S}\left(t|\mathbf{x}_{i}^{*}\right) & =\frac{\sum_{k=1}^{K}\bar{w}_{J}^{k}\exp\left(-\left\{ \sum_{j=1}^{h-1}\left(\tau_{j}-\tau_{j-1}\right)\exp\left(\mathbf{z}_{i}^{*}{}^{\prime}\boldsymbol{\beta}_{j}^{k}\right)\right\} -\left(t-\tau_{h-1}\right)\exp\left(\mathbf{z}_{i}^{*}{}^{\prime}\boldsymbol{\beta}_{h}^{k}\right)\right)}{\sum_{k=1}^{K}\bar{w}_{J}^{k}},\label{eq: predicted survival}\end{aligned}$$ if $\tau_{h-1}\leq t_{i}<\tau_{h},\,h\leq J$.

Applications\[sec:Applications\]
================================

Simulations\[subsec:Simulations\]
---------------------------------

A simulation study is conducted in order to assess the performance of PSLiB in various senarios and compare it with the state-of-the-art MCMC aproach in @wagner2011bayesian. The censoring indicators $d_{i}$ ($i=1,\ldots,n$) are simulated from a Bernoulli distribution with probability $1-p_{c}$ ($p_{c}$ being the proportion of censored observations), and the 