Higher-order Common Information

Shannon’s mutual information quantifies redundancy between two random variables. We introduce a new notion, termed higher-order common information (HCI), which captures the information shared among $n$ arbitrarily distributed random variables. The quantity is defined through an iterative information-bottleneck construction and can be interpreted as the maximum rate at which a single compressed representation can simultaneously preserve information about all variables. For jointly Gaussian and Bernoulli sources, we derive closed-form expressions for any $n$. We furthermore show that the HCI yields strictly tighter characterizations of redundancy than existing bounds, and demonstrate how to numerically approximate the HCI for arbitrarily distributed sources.