On the Conditional Distributions of Spatial Point Processes

Abstract : We consider the problem of estimating a latent point process, given the realization of another point process on abstract measurable state spaces. First, we establish an expression of the conditional distribution of a latent Poisson point process given the observation process when the transformation from the latent process to the observed process includes displacement, thinning and augmentation with extra points. We present an original analysis based on a self-contained random measure theoretic approach combined with reversed Markov kernel techniques. This simplifies and complements previous derivations given in \cite{maler}, \cite{zuev}. Second, we show how to extend our analysis to the more complicated case where the latent point process is associated to triangular array sequences, yielding what seems to be the first results of this type for this class of spatial point processes.