MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Quantifying unique information

Nils Bertschinger, Johannes Rauh, Eckehard Olbrich, Jürgen Jost and Nihat Ay


We propose new measures of shared information, unique information and synergistic information that can be used to decompose the multi-information of a pair of random variables $(Y,Z)$ with a third random variable $X$. Our measures are motivated by an operational idea of unique information which suggests that shared information and unique information should depend only on the pair marginal distributions of $(X,Y)$ and $(X,Z)$. Although this invariance property has not been studied before, it is satisfied by other proposed measures of shared information. The invariance property does not uniquely determine our new measures, but it implies that the functions that we define are bounds to any other measures satisfying the same invariance property. We study properties of our measures and compare them to other candidate measures.

MSC Codes:
94A15, 94A17
Shannon information, mutual information, information decomposition, synergy

Related publications

2014 Journal Open Access
Nils Bertschinger, Johannes Rauh, Eckehard Olbrich, Jürgen Jost and Nihat Ay

Quantifying unique information

In: Entropy, 16 (2014) 4, pp. 2161-2183