Computes the Watanabe-Akaike Information Criterion (WAIC) for a fitted `exnex_surv` model. WAIC is a fully Bayesian information criterion that uses the posterior draws of the log-likelihood and adds a penalty for effective number of parameters \(p_{waic}\).
Value
A named list with elements `waic`, `se_elpd_waic`, `lpd`, `p_waic`, `elpd_waic`, and `pointwise` (an `n` by 3 matrix with columns `lpd`, `p_waic`, `waic`).
Details
For each observation \(i\), with posterior draws indexed by \(s=1,\ldots,S\), we compute the pointwise log-likelihood \(\log p(y_i \mid \theta^{(s)})\), where the log-normal AFT model gives for an observed event (\(\delta_i=1\)) $$\log p(y_i \mid \theta^{(s)}) = \log\phi\!\left(\frac{\log t_i-\eta_i^{(s)}}{\sigma^{(s)}}\right) - \log(\sigma^{(s)} t_i),$$ and for a censored observation (\(\delta_i=0\)) $$\log p(y_i \mid \theta^{(s)}) = \log\!\left[1-\Phi\!\left(\frac{\log t_i-\eta_i^{(s)}}{\sigma^{(s)}}\right)\right],$$ where \(\eta_i^{(s)}\) is the linear predictor built from the draws.
WAIC is then $$lpd = \sum_i \log\!\left(\frac{1}{S}\sum_s \exp\log p(y_i\mid\theta^{(s)})\right),$$ $$p_{waic} = \sum_i \mathrm{var}_s\big(\log p(y_i\mid\theta^{(s)})\big),$$ $$\mathrm{WAIC} = -2(lpd - p_{waic}).$$ Rows with extreme leverage (`p_waic` above a large threshold) may flag issues; the function returns a `pointwise` matrix so users can investigate.