Poisson processes are used in various application fields applications (public health biology, reliability and so on). In their homogeneous version, the intensity process is a deterministic constant. In their inhomogeneous version, it depends on time. To allow for an endogenous evolution of the intensity process we consider multiplicative intensity processes. Inference methods have been developed when the trajectories are fully observed. We deal with the case of a partially observed process. As a motivating example, consider the analysis of an electrical network through time. This network is composed of cables and accessories (joints). When a cable fails, the cable is replaced by a new cable connected to the network by two new accessories. When an accessory fails, the same kind of reparation is done leading to the addition of only one accessory. The failure rate depends on the stochastically evolving number of accessories. We only observe the times events; the initial number of accessories and the cause of the incident (cable or accessory) are only partially observed. The aim is to estimate the different failure rates or to make predictions. The inference is strongly influenced by the initial number of accessories, which is typically an unknown quantity. We deduce a sensible prior on the initial number of accessories using the probabilistic properties of the process . We illustrate the performances of our methodology on a large simulation study.
