  Bayesian paradigm takes advantage of well fitting complicated survival models
and feasible computing in survival analysis owing to the superiority in
tackling the complex censoring scheme, compared with the frequentist paradigm.
In this chapter, we aim to display the latest tendency in Bayesian computing,
in the sense of automating the posterior sampling, through Bayesian analysis of
survival modeling for multivariate survival outcomes with complicated data
structure. Motivated by relaxing the strong assumption of proportionality and
the restriction of a common baseline population, we propose a generalized
shared frailty model which includes both parametric and nonparametric frailty
random effects so as to incorporate both treatment-wise and temporal variation
for multiple events. We develop a survival-function version of ANOVA dependent
Dirichlet process to model the dependency among the baseline survival
functions. The posterior sampling is implemented by the No-U-Turn sampler in
Stan, a contemporary Bayesian computing tool, automatically. The proposed model
is validated by analysis of the bladder cancer recurrences data. The estimation
is consistent with existing results. Our model and Bayesian inference provide
evidence that the Bayesian paradigm fosters complex modeling and feasible
computing in survival analysis and Stan relaxes the posterior inference.
