Abstract
A class of random hazard rates, which is defined as a mixture of an indicator kernel convoluted 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 $\mathbf{S}-paths. A closed and tractable Bayes estimator for the hazard rate is derived to be a finite sum over $\mathbf{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 $\mathbf{S}-path in Bayesian modeling of monotone hazard rates. An efficient Markov chain Monte Carlo method is proposed to approximate this class of estimates. It is shown that $\mathbf{S}-path characterization also exists in modeling with covariates by a proportional hazard model, and the proposed algorithm again applies.