A non-homogeneous count process: marginalizing a Poisson driven Cox process

The paper considers a Cox process where the stochastic intensity function for the Poisson data model is itself a non-homogeneous Poisson process. We show that it is possible to obtain the marginal data process, namely a non-homogeneous count process exhibiting over-dispersion. The model generates intensity functions which are non-decreasing. This is not a restriction in practice since observed data can always be transformed which guarantees a non-decreasing intensity function. We are able, for a specific choice of Cox process, to mathematically marginalize out the latent process leaving with a directly available likelihood function. This makes inference much simpler compared to algorithms which retain the latent process.