LAUSR

This is an excellent and extremely well-written summary of recent research on self-exciting spatialtemporal point processes. It contributes very nicely to the literature and I will use it personally to teach my graduate students about the topic. The author should be congratulated for his excellent writing. I would like to comment briefly on estimation. Again, Reinhart provides a superb review, and seeing the current state of knowledge regarding maximum likelihood estimation (MLE) and its variants, one may walk away from this article with the misleading impression that estimation for spatial-temporal point processes is a solved problem that can readily be attacked not only by MLE but also by various other techniques such as E-M or stochastic reconstruction. However, in practice there are real problems with the implementation of many of the methods here. The first and in my opinion main shortcoming of MLE is the integral term in Reinhart’s equation (8). For some very simple models, this integral can be computed numerically as a function of the parameters being estimated, but this is rare. In practice, one must approximate this integral numerically. The problem is that, in MLE, one is searching over a vast parameter space, and the numerical approximation to the integral must be a close approximation for all of the parameter space, or else the optimization function may choose some parameter vector where the approximation is poor. Anyone who has dealt with MLE knows the sort of Murphy’s Law to which I am referring. If anything can possibly go wrong with the approximation to the likelihood function, MLE seems to have a way of gravitating to it. Harte (2013) comments nicely on the importance of the issue of integral approximation in MLE in practice. Another issue with the integral