A graduate student came to see me the other day with a bivariate Poisson distribution and a question about using EM in this framework. The problem boils down to adding one correlation parameter and an extra term in the likelihood
Both terms involving sums are easy to deal with, using latent variables as in mixture models. The subtractions are trickier, as the negative parts cannot appear in a conditional distribution. Even though the problem can be handled by a direct numerical maximisation or by an almost standard Metropolis-within-Gibbs sampler, my suggestion regarding EM per se was to proceed by conditional EM, one parameter at a time. For instance, when considering conditional on both Poisson parameters, depending on whether or not, one can consider either
thus producing a Beta-like target function in after completion, or turn
to produce a Beta-like target function in after completion. In the end, this is a rather pedestrian exercise and I am still frustrated at missing the trick to handle the subtractions directly, however it was nonetheless a nice question!