Higher-order Common Information

We present a new notion R_ℓ of higher-order common information, which quantifies the information that ℓ≥ 2 arbitrarily distributed random variables have in common. We provide analytical lower bounds on R_3 and R_4 for jointly Gaussian distributed sources and provide computable lower bounds for R_ℓ for any ℓ and any sources. We also provide a practical method to estimate the lower bounds on, e.g., real-world time-series data. As an example, we consider EEG data acquired in a setup with competing acoustic stimuli. We demonstrate that R_3 has descriptive properties that is not in R_2. Moreover, we observe a linear relationship between the amount of common information R_3 communicated from the acoustic stimuli and to the brain and the corresponding cortical activity in terms of neural tracking of the envelopes of the stimuli.