We study the joint occurrence of large values of a Markov random field or undirected graphical model associated to a block graph. On such graphs, containing trees as special cases, we aim to generalize recent results for extremes of Markov trees. Every pair of nodes in a block graph is connected by a unique shortest path. These paths are shown to determine the limiting distribution of the properly rescaled random field given that a fixed variable exceeds a high threshold. The latter limit relation implies that the random field is multivariate regularly varying and it determines the max-stable distribution to which component-wise maxima of independent random samples from the field are attracted. When the sub-vectors induced by the blocks have certain limits parametrized by H\"usler-Reiss distributions, the global Markov property of the original field induces a particular structure on the parameter matrix of the limiting max-stable H\"usler-Reiss distribution. The multivariate Pareto version of the latter turns out to be an extremal graphical model according to the original block graph. Thanks to these algebraic relations, the parameters are still identifiable even if some variables are latent.

翻译：我们研究了与块图相关的马尔可夫随机场或无向图模型中大值的联合发生。在这类包含树作为特例的图上，我们旨在推广马尔可夫树极值的现有结果。块图中的每一对节点均由唯一的最短路径连接。这些路径被证明能够决定在给定固定变量超过高阈值条件下，经适当重标度后的随机场的极限分布。该极限关系表明，随机场是多变量正则变化的，并且决定了场中独立随机样本的逐分量最大值所吸引的最大稳定分布。当由块诱导的子向量具有由Hüsler-Reiss分布参数化的特定极限时，原场的全局马尔可夫性质会在极限最大稳定Hüsler-Reiss分布的参数矩阵上引入特定结构。该分布的多元帕累托版本被证明是根据原始块图的极值图模型。借助这些代数关系，即使某些变量是潜在的，参数仍然可识别。