  Finite sample inference for Cox models is an important problem in many
settings, such as clinical trials. Bayesian procedures provide a means for
finite sample inference and incorporation of prior information if MCMC
algorithms and posteriors are well behaved. On the other hand, estimation
procedures should also retain inferential properties in high dimensional
settings. In addition, estimation procedures should be able to incorporate
constraints and multilevel modeling such as cure models and frailty models in a
straightforward manner. In order to tackle these modeling challenges, we
propose a uniformly ergodic Gibbs sampler for a broad class of convex set
constrained multilevel Cox models. We develop two key strategies. First, we
exploit a connection between Cox models and negative binomial processes through
the Poisson process to reduce Bayesian computation to iterative Gaussian
sampling. Next, we appeal to sufficient dimension reduction to address the
difficult computation of nonparametric baseline hazards, allowing for the
collapse of the Markov transition operator within the Gibbs sampler based on
sufficient statistics. We demonstrate our approach using open source data and
simulations.
