State estimation for cox processes on general spaces

Let N be an observable Cox process on a locally compact space E directed by an unobservable random measure M. Techniques are presented for estimation of M, using the observations of N to calculate conditional expectations of the form E M]|$\text{F}$_A], where $\text{F}$_A is the σ–algebra generated by the restriction of N to A. We introduce a random measure whose distribution depends on N_A, from which we obtain both exact estimates and a recursive method for updating them as further observations become available. Application is made to the specific cases of estimation of an unknown, random scalar multiplier of a known measure, of a symmetrically distributed directing measure M and of a Markov–directed Cox process on $\text{R}$. By means of a Poisson cluster representation, the results are extended to treat the situation where N is conditionally additive and infinitely divisible given M.