Title: Exact partial information decompositions for Gaussian systems based on dependency constraints,
Abstract: The Partial Information Decomposition (PID) [arXiv:1004.2515] provides a
theoretical framework to characterize and quantify the structure of
multivariate information sharing. A new method (Idep) has recently been
proposed for computing a two-predictor PID over discrete spaces.
[arXiv:1709.06653] A lattice of maximum entropy probability models is
constructed based on marginal dependency constraints, and the unique
information that a particular predictor has about the target is defined as the
minimum increase in joint predictor-target mutual information when that
particular predictor-target marginal dependency is constrained. Here, we apply
the Idep approach to Gaussian systems, for which the marginally constrained
maximum entropy models are Gaussian graphical models. Closed form solutions for
the Idep PID are derived for both univariate and multivariate Gaussian systems.
Numerical and graphical illustrations are provided, together with practical and
theoretical comparisons of the Idep PID with the minimum mutual information PID
(Immi). [arXiv:1411.2832] In particular, it is proved that the Immi method
generally produces larger estimates of redundancy and synergy than does the
Idep method. In discussion of the practical examples, the PIDs are complemented
by the use of deviance tests for the comparison of Gaussian graphical models.