The entropic region is formed by the collection of the Shannon entropies of all subvectors of finitely many jointly distributed discrete random variables. For four or more variables, the structure of the entropic region is mostly unknown. We utilize a variant of the Maximum Entropy Method to obtain five-variable non_shannon entropy inequalities, which delimit the five-variable entropy region. This method adds copies of some of the random variables in generations. A significant reduction in computational complexity, achieved through theoretical considerations and by harnessing the inherent symmetries, allowed us to calculate all five-variable non-Shannon inequalities provided by the first nine generations. Based on the results, we define two infinite collections of such inequalities and prove them to be entropy inequalities. We investigate downward-closed subsets of non-negative lattice points that parameterize these collections, and based on this, we develop an algorithm to enumerate all extremal inequalities. The discovered set of entropy inequalities is conjectured to characterize the applied method completely.

翻译：熵区域由有限多个联合分布的离散随机变量的所有子向量的香农熵集合构成。对于四个或更多变量，熵区域的结构大多未知。我们利用最大熵方法的一种变体来获得五变量非香农熵不等式，这些不等式界定了五变量熵区域。该方法通过在迭代过程中添加某些随机变量的副本。通过理论考量并利用内在对称性实现的显著计算复杂度降低，使我们能够计算前九次迭代提供的所有五变量非香农不等式。基于这些结果，我们定义了两个此类不等式的无限集合，并证明它们是熵不等式。我们研究了参数化这些集合的非负格点的向下封闭子集，并在此基础上开发了一种枚举所有极值不等式的算法。所发现的熵不等式集合被推测能完全刻画所应用方法的特征。