Conditional and relevant common information

Two variations on Wyner's common information are proposed: conditional common information and relevant common information. The former characterizes the minimum common rate that is required for lossless source-coding over a one-to-two Gray-Wyner network, when the sum-rate is restricted to be minimal and the terminals all share the side-information. It also characterizes the minimum rate of common randomness that is required for two terminals sharing some side-information to strongly coordinate their outputs according to a target distribution. The latter, relevant common information, is an upper bound on the minimum common rate required for two receivers of a one-to-two Gray-Wyner network to weakly coordinate their reconstruction sequences with the source according to a target distribution. It also characterizes the minimum rate of common randomness that is required for two terminals to produce a target strongly-coordinated sequence at the output of a two-user multiple-access channel.