This paper is concerned with nonparametric Bayesian inference of
the Aalen’s multiplicative counting process model. For a desired
nonparametric prior distribution of the cumulative intensity function, a class
of Lévy processes is considered, and it is shown that the class of
Lévy processes is conjugate for the multiplicative counting process
model, and formulas for obtaining a posterior process are derived. Finally, our
results are applied to several practically important models such as one point
processes for right-censored data, Poisson processes and Markov processes.