Maximum likelihood estimation for the bombing model

Perhaps the best known example of a random set is the Boolean model. It is the union of `grains' such as discs, squares or triangles which are placed at the points of a Poisson point process. The Poisson points are called the `germs'. We are interested in estimating the intensity, say lambda, of the Poisson process from a sample of a Boolean model of discs (the bombing model). A natural estimate is the number of germs in the observation region divided by the area of that region. Unfortunately, we do not observe the presence of a given germ when its associated disc is completely covered by other discs. On the other hand, we observe the exact location of a germ when we observe any part of its associated disc's boundary. We demonstrate how to apply Coupling From The Past to sample from the conditional distribution, given the data, of the unobserved germs. Such samples allow us to approximate the maximum likelihood estimator of the intensity. We discuss and compare two methods to do so. The first method is based on a Monte Carlo approximation of the likelihood function. The second is a stochastic version of the EM algorithm.