Partial monitoring of multistate systems

For large multicomponent systems it is typically too costly to monitor the entire system constantly. In the present paper we consider a case where a component is unobserved in a time interval [0, T]. The time T is a stochastic variable with a distribution which depends on the structure of the system and the lifetime distribution of the other components. Different systems will result in different distributions of T. The main focus is on how the unobserved period of time affects what we learn about the unobserved component during this period. We analyse this by considering one single component in three different cases. In the first case we consider both T as well as the state of the unobserved component at time T as given. In the second case we allow the state of the unobserved component at time T to be stochastic, while in the third case both T and the state are treated as stochastic variables. In all cases we study the problem using preposterior analysis. That is, we investigate how much information we can expect to get by the end of the time interval [0, T]. The methodology is also illustrated on a more complex example.