Multiple event timedata (e.g., carcinogenic growths in different times and locations, multiple attacks of cardiac arrest) arise in various medical studies. A Bayesian analysis of such data based onproportional intensity modelof multiple event time data is presented in this paper. The Bayesian structure is somewhat analogous to that used by Kalbfleisch in a proportional hazard model. An unobserved randomfrailtycomponent is used in the proportional intensity model to take care of heterogeneity among the intensity processes in different subjects. The Monte Carlo method of sampling from multivariate distributions, the so-called Gibbs sampler, is used to sample from the joint posterior distribution of the unknown parameters. The methodology developed here is exemplified with the well-known data set on rat tumors of Gail, Santner, and Brown.