A rational probability distribution on four binary random variables $X, Y, Z, U$ is constructed which satisfies the conditional independence relations $[X \mathrel{\text{$\perp\mkern-10mu\perp$}} Y]$, $[X \mathrel{\text{$\perp\mkern-10mu\perp$}} Z \mid U]$, $[Y \mathrel{\text{$\perp\mkern-10mu\perp$}} U \mid Z]$ and $[Z \mathrel{\text{$\perp\mkern-10mu\perp$}} U \mid XY]$ but whose entropy vector violates the Ingleton inequality. This settles a recent question of Studen\'y (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. The last case in the classification of which of these inequalities are essentially conditional is also settled.

翻译：构造了一个定义在四个二值随机变量 $X, Y, Z, U$ 上的有理概率分布，该分布满足条件独立关系 $[X \mathrel{\text{$\perp\mkern-10mu\perp$}} Y]$、$[X \mathrel{\text{$\perp\mkern-10mu\perp$}} Z \mid U]$、$[Y \mathrel{\text{$\perp\mkern-10mu\perp$}} U \mid Z]$ 和 $[Z \mathrel{\text{$\perp\mkern-10mu\perp$}} U \mid XY]$，但其熵向量违反了英格顿不等式。这解决了 Studený 近期提出的一个问题（IEEE Trans. Inf. Theory, vol. 67, no. 11），并表明在四个离散随机变量的条件独立假设中，恰好存在十个（考虑对称性下的）包含关系极小集，使得英格顿不等式成立。此外，关于哪些不等式本质上是条件型的分类中的最后一个情况也得以解决。