  A class of random hazard rates, which is defined as a mixture of an indicator
kernel convolved with a completely random measure, is of interest. We provide
an explicit characterization of the posterior distribution of this mixture
hazard rate model via a finite mixture of S-paths. A closed and tractable Bayes
estimator for the hazard rate is derived to be a finite sum over S-paths. The
path characterization or the estimator is proved to be a Rao--Blackwellization
of an existing partition characterization or partition-sum estimator. This
accentuates the importance of S-paths in Bayesian modeling of monotone hazard
rates. An efficient Markov chain Monte Carlo (MCMC) method is proposed to
approximate this class of estimates. It is shown that S-path characterization
also exists in modeling with covariates by a proportional hazard model, and the
proposed algorithm again applies. Numerical results of the method are given to
demonstrate its practicality and effectiveness.
