论文标题
没有第11条条件的Ingleton不等式
No eleventh conditional Ingleton inequality
论文作者
论文摘要
构建了四个二进制随机变量$ x，y，z，u $上的合理概率分布，该变量满足有条件的独立关系$ [x \ mathrel {\ text {$ \ perp \ perp \ perp \ mkern-10mu \ perp $}} y] $， \ mid U] $，$ [y \ MATHREL {\ text {$ \ perp \ perp \ mkern-10mu \ perp $}} u \ mid z] $和$ [z \ mathrel {\ mathrel {\ text {$ \ perp \ perp \ perp \ mkern-10mu \ perp $}}这解决了最近的一个问题（IEEETrans。Inf。理论第67卷，第11号），并表明，在对称性的情况下，对于四个离散的随机变量，确切的有条件独立性假设恰好是十个纳入的纳入最低限度集合，这些变量使Ingleton不等式持有。这些不平等基本上有条件的分类中的最后一个情况也解决了。
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 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. The last case in the classification of which of these inequalities are essentially conditional is also settled.
