Conditional and Relevant Common Information

Two variations on Wyner's common information are proposed: conditional common information and relevant common information. These are shown to have operational meanings analogous to those of Wyner's common information in appropriately defined distributed problems of compression, simulation and channel synthesis. For relevant common information, an additional operational meaning is identified: on a multiple-access channel with private and common messages, it is the minimal common-message rate that enables communication at the maximum sum-rate under a weak coordination constraint on the inputs and output. En route, the weak-coordination problem over a Gray-Wyner network is solved under the no-excess-rate constraint.