State estimation in systems driven by poisson processes with unknown arrival rates

The Poisson process is a very useful tool for modeling system inputs or measurement disturbances in a wide variety of applications like tracking problems, failing systems, etc. This paper investigates the problem of state estimation in systems driven by or observed in the presence of Poisson noise whose arrival rate is unknown but parametrically constrained to be an element of a known compact space. This includes convex uncertainty spaces and covers a wide range of applications since most uncertainties can be restricted to such a set on physical grounds. It is demonstrated that a combined detection estimation structure and an incremental square error criterion provide a minimax solution which is not pessimistic for rates far from the least favorable prior, is straightforward to realize and capable of adaptation to changing rates. Applications in Poisson driven linear systems and doubly stochastic Poisson processes are considered. It is worthy of note that the resulting solution performs nearly uniformly close to the MMSE solution obtained if the real rate were known without requiring any prior probability assumptions.