Bayes derivation of multitarget intensity filters

The multitarget intensity filter is derived from a Bayesian first principles approach using a Poisson point process approximation at one step. The prior multitarget model is assumed to be a Poisson point process. The Bayes multitarget posterior probability density function is first defined on the Poisson event space, and then reformulated in terms of the intensity functions that characterize all Poisson point processes. It is shown that the predicted multitarget and predicted measurement processes are Poisson. However, the multitarget Bayes posterior probability density is not that of a Poisson point process. It is shown that all the single-target marginal probability density functions of the multitarget posterior probability density are identical. Consequently, the multitarget Bayes posterior probability density is approximated as the product of its marginal probability densities. Maximum likelihood determines the scale factor that converts the marginal probability density to a posterior multitarget intensity. This posterior multitarget intensity defines the approximating information updated multitarget Poisson point process and is very similar to the intensity function produced by the PHD filter.