Exact sampling from conditional Boolean models with applications to maximum likelihood inference (Q2749125)

The Boolean model of discs (the bombing model) is the union of unit balls (grains) centred at the points of a stationary Poisson point process (germs). The authors attack the estimation problem for its intensity using the maximum likelihood approach, while most of previous approaches relied on variants of the method of moments [see \textit{I. Molchanov}, Statistics of the Boolean model for practitioners and mathematicians. (1997; Zbl 0878.62068)]. NEWLINENEWLINENEWLINEBecause the balls may overlap, form clusters and so render some grains completely invisible, it is not possible to come up with an explicit expression for the likelihood. The authors treat it as missing data problem and demonstrate how coupling from the past can be applied to generate samples from the conditional distributions of germs given the observed union set. These samples are then used to approximate the maximum likelihood estimator of the intensity. The authors discuss and compare two methods: one based on a Monte Carlo approximation of the likelihood function, the other a stochastic version of the EM algorithm.