Filtering the histories of a partially observed marked point process

We consider a situation in which the evolution of an ‘underlying’ marked point process is of interest, but where this process is not directly observable. Instead, we assume that another marked point process, which is fully determined by the underlying process, can be observed. The problem is then the estimation, at any given time t, of the underlying development so far, given the corresponding observations. The solution, in the sense of a conditional distribution of the underlying pre-t history, is shown to satisfy a recursive filter formula. Sufficient conditions for the uniqueness of the solution are given. Two non-trivial examples are considered in detail.