On the relationships between lumpability and filtering of finite stochastic systems

The aim of this paper is to provide the conditions necessary to reduce the complexity of state filtering for finite stochastic systems (FSSs). A concept of lumpability for FSSs is introduced. In this paper we assert that the unnormalised filter for a lumped FSS has linear dynamics. Two sufficient conditions for such a lumpability property to hold are discussed. We show that the first condition is also necessary for the lumped FSS to have linear dynamics. Next, we prove that the second condition allows the filter of the original FSS to be obtained directly from the filter for the lumped FSS. Finally, we generalise an earlier published result for the approximation of a general FSS by a lumpable FSS.