Nonparametric Bayes estimator of survival functions for doubly/interval censored data

The non-parametric Bayes estimator with Dirichlet process prior of a survival function based on right censored data was considered by Susarla and Van Ryzin (1976) and many others. We obtain the non-parametric Bayes estimator of a survival function when data are right, left or interval censored. The resulting Bayes estimator with Dirichlet process prior has an explicit formula. In contrast, there is no explicit formula known for the non-parametric maximum likelihood estimator (NPMLE) with such data. In fact, we show that the NPMLE with doubly/interval censored data cannot, in general, be the limit of Bayes estimators for any sequence of priors. Several examples are given, showing that the NPMLE and the non-parametric Bayes estimator may or may not be the same, even when the prior is 'non-informative'.