Bayesian Inference for Survival Data with Nonparametric Hazards and Vague

Statistical inference is reviewed for survival data applications with hazard models having one parameter per distinct failure time and using Jeffreys' (1961) vague priors. Distinction between a discrete hazard and a piecewise exponential model is made. Bayes estimators of survival probabilities ace derived. For a single sample and a discrete hazard, the Bayes estimator is shown to be larger than Nelson's (1972) which in turn is larger than KaplanMeier's (1958) estimator. With a piecewise exponential model, the Bayes estimator is also shown to be larger than that using maximum likelihood. Presuming a proportional hazards formulation to incorporate covariate information and a discrete underlying hazard model, the marginal posterior distribution of the regression parameters is proportional to Breslow's (1974) approximation to the marginal likelihood of Kalbfleisch and Prentice (1973). A refinement of Breslow's (1974) approximate likelihood is obtained when a piecewise exponential model is used for the underlying hazard. These results serve as illustrations of differences between estimators obtained from a frequentist's approach and a Bayes strategy with vague priors. Further, the Bayes results have practical advantages.