We develop a Bayesian model for continuous-time incurred but not yet reported (IBNYR) events under four types of secondary data, and show that unreported events, such as claims, have a Poisson distribution with a reduced arrival parameter if event arrivals are Poisson distributed. Using insurance claims as an example of an IBNYR event, we apply Markov chain Monte Carlo (MCMC) to the continuous-time IBNYR claims model of Jewell using Type I and Type IV data. We illustrate the relative stability of the MCMC method versus the Gammoid approximation of Jewell by showing that the MCMC estimates approach their prior parameters, while the Gammoid approximations grow without bound for Type IV data. Moreover, this holds for any distribution that the delay parameter is assumed to follow. Our framework also allows for the computation of posterior confidence intervals for the parameters.