No eleventh conditional Ingleton inequality

04/08/2022

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by   Tobias Boege, et al.

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A rational probability distribution on four binary random variables X, Y, Z, U is constructed which satisfies the conditional independence relations [X ⊥-10mu⊥ Y], [X ⊥-10mu⊥ Z | U], [Y ⊥-10mu⊥ U | Z] and [Z ⊥-10mu⊥ U | XY] but whose entropy vector violates the Ingleton inequality. This settles a recent question of Studený (IEEE Trans. Inf. Theory vol. 67, no. 11) and shows that there are, up to symmetry, precisely ten inclusion-minimal sets of conditional independence assumptions on four discrete random variables which make the Ingleton inequality hold.