Management of aging engineering systems is a growing concern. Decisions such as service limitation or interruption might have to be taken in order to keep an appropriate level of safety following a disastrous event. Information about the infrastructure can be collected by inspections and/or by Structural Health Monitoring (SHM) devices. The Value of Information (VoI) from Bayesian decision analysis can be used to quantify the benefit of using a certain data-acquisition strategy and to select the most appropriate one. This paper takes advantage of the correlation among the state of the components of an infrastructure due, for example, to their similar design and exposure conditions. The collection of data about an appropriately selected sample of components is used to infer the state of other components. As an example, the proposed formulation is used to quantify the VoI considering a system of bridges subject to gradual degradation processes and seismic hazards.