I have a point process defined in two parts as follows. Consider first the main process which we call $A$ which is homogeneous Poisson process with conditional intensity

$$\lambda(t) = \mu$$

For every arrival time $x_i$ that $A$ produces we run another process which we call $B$. The conditional intensity of this process assuming $t\geq x_i$ is

$$\nu(t) = \gamma e^{-(t-x_i)}$$

The combined process simply combines the list of arrival times from $A$ and $B$.

This is easy enough to simulate if you are given the parameters $\mu$ and $\gamma$.

Given some real data listing arrival times, how can one compute the MLE for the two parameters of the combined point process in an efficient way?

There seems to be a combinatorial explosion that I can't see how to avoid. I understand there is unlikely to be a closed form solution.

To give some more details of the difficulty in doing this. If you know which points are caused by process $A$ then you can compute the likelihood and use an optimization procedure to find the parameters $\mu$ and $\gamma$ which give the MLE. The problem is that you don't know that and trying all $2^n$ possibilities (where $n$ is the number of points in the data) is too much. It is possible that something from the HMM training literature might help but I am not expert enough in that to be able to tell.

For a homogeneous Poisson process alone on a time interval $(0,T)$ which has $n$ points, the MLE is of course $n/T$.

Cross-posted to https://cstheory.stackexchange.com/questions/25295/an-efficient-method-to-find-the-mle-of-the-combination-of-two-point-processes