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Exact Partial Information Decompositions for Gaussian Systems Based on Dependency Constraints

1
Department of Statistics, University of Glasgow, Glasgow G12 8QQ, UK
2
Institute of Neuroscience and Psychology, University of Glasgow, Glasgow G12 8QQ, UK
*
Author to whom correspondence should be addressed.
Entropy 2018, 20(4), 240; https://doi.org/10.3390/e20040240
Received: 9 March 2018 / Revised: 26 March 2018 / Accepted: 27 March 2018 / Published: 30 March 2018
(This article belongs to the Section Information Theory, Probability and Statistics)
The Partial Information Decomposition, introduced by Williams P. L. et al. (2010), provides a theoretical framework to characterize and quantify the structure of multivariate information sharing. A new method ( I dep ) has recently been proposed by James R. G. et al. (2017) for computing a two-predictor partial information decomposition over discrete spaces. 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 I dep approach to Gaussian systems, for which the marginally constrained maximum entropy models are Gaussian graphical models. Closed form solutions for the I dep PID are derived for both univariate and multivariate Gaussian systems. Numerical and graphical illustrations are provided, together with practical and theoretical comparisons of the I dep PID with the minimum mutual information partial information decomposition ( I mmi ), which was discussed by Barrett A. B. (2015). The results obtained using I dep appear to be more intuitive than those given with other methods, such as I mmi , in which the redundant and unique information components are constrained to depend only on the predictor-target marginal distributions. In particular, it is proved that the I mmi method generally produces larger estimates of redundancy and synergy than does the I dep method. In discussion of the practical examples, the PIDs are complemented by the use of tests of deviance for the comparison of Gaussian graphical models. View Full-Text
Keywords: partial information decomposition; mutual information; unique information; dependency constraints; Gaussian graphical models; maximum entropy partial information decomposition; mutual information; unique information; dependency constraints; Gaussian graphical models; maximum entropy
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Kay, J.W.; Ince, R.A.A. Exact Partial Information Decompositions for Gaussian Systems Based on Dependency Constraints. Entropy 2018, 20, 240.

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