Given a fixed dependency graph G that describes a Bayesian network of binary variables X1,…,Xn, our main result is a tight bound on the mutual information Ic(Y1,…,Yk)=∑kj=1H(Yj)/c−H(Y1,…,Yk) of an observed subset Y1,…,Yk of the variables X1,…,Xn. Our bound depends on certain quantities that can be computed from the connective structure of the nodes in G. Thus it allows to discriminate between different dependency graphs for a probability distribution, as we show from numerical experiments.
In this paper, we explore a connection between binary hierarchical models, their marginal polytopes, and codeword polytopes, the convex hulls of linear codes. The class of linear codes that are realizable by hierarchical models is determined. We classify all full dimensional polytopes with the property that their vertices form a linear code and give an algorithm that determines them.
We offer a new approach to the \emph{information decomposition} problem in information theory: given a `target' random variable co-distributed with multiple `source' variables, how can we decompose the mutual information into a sum of non-negative terms that quantify the contributions of each random variable, not only individually but also in combination? We define a new way to decompose the mutual information, which we call the \emph{Information Attribution} (IA), and derive a solution using cooperative game theory. It can be seen as assigning a "fair share'' of the mutual information to each combination of the source variables. Our decomposition is based on a different lattice from the usual `partial information decomposition' (PID) approach, and as a consequence {the IA} has a smaller number of terms {than PID}: it has analogs of the synergy and unique information terms, but lacks separate terms corresponding to redundancy, instead sharing redundant information between the unique information terms. Because of this, it is able to obey equivalents of the axioms known as `local positivity' and `identity', which cannot be simultaneously satisfied by a PID measure., Nihat Ay, Daniel Polani and Nathaniel Virgo., and Obsahuje bibliografické odkazy
Stochastic interdependence of a probability distribution on a product space is measured by its Kullback-Leibler distance from the exponential family of product distributions (called multi-information). Here we investigate low-dimensional exponential families that contain the maximizers of stochastic interdependence in their closure. Based on a detailed description of the structure of probability distributions with globally maximal multi-information we obtain our main result: The exponential family of pure pair-interactions contains all global maximizers of the multi-information in its closure.
We investigate solution sets of a special kind of linear inequality systems. In particular, we derive characterizations of these sets in terms of minimal solution sets. The studied inequalities emerge as information inequalities in the context of Bayesian networks. This allows to deduce structural properties of Bayesian networks, which is important within causal inference.