The Hawkes process is a practically and theoretically important class
of point processes, but parameter-estimation for such a process can pose
various problems. In this paper we explore and compare two approaches
to Bayesian inference. The first approach is based on the so-called conditional
intensity function, while the second approach is based on an underlying
clustering and branching structure in the Hawkes process. For
practical use, MCMC (Markov chain Monte Carlo) methods are employed.
The two approaches are compared numerically using three examples of the
Hawkes process.