Graphical models with heavy-tailed factors can be used to model extremal dependence or causality between extreme events. In a Bayesian network, variables are recursively defined in terms of their parents according to a directed acyclic graph (DAG). We focus on max-linear graphical models with respect to a special type of graphs, which we call a \emph{tree of transitive tournaments}. The latter are block graphs combining in a tree-like structure a finite number of transitive tournaments, each of which is a DAG in which every two nodes are connected. We study the limit of the joint tails of the max-linear model conditionally on the event that a given variable exceeds a high threshold. Under a suitable condition, the limiting distribution involves the factorization into independent increments along the shortest trail between two variables, thereby imitating the behavior of a Markov random field. We are also interested in the identifiability of the model parameters in case some variables are latent and only a subvector is observed. It turns out that the parameters are identifiable under a criterion on the nodes carrying the latent variables which is easy and quick to check.
翻译:具有重尾因素的图形模型可以用来模拟极端事件之间的极端依赖性或因果关系。 在巴伊西亚网络中, 变量会根据定向环状图( DAG) 以父母为对象的递归性定义。 我们注重于特定类型的图形的最大线性图形模型, 我们称之为“ 过渡性锦标之树 ” 。 后者是在树形结构中结合一定数量的过渡性锦标赛的区块图, 每个赛都是连接两个节点的DAG 。 我们研究最大线性模型联合尾部的极限, 条件是某个变量超过一个高阈值。 在一个合适的条件下, 限制分布意味着在两个变量之间最短的轨迹上将因素化为独立的递增值, 从而模仿Markov 随机场的行为。 我们还对模型参数在有些变量具有潜伏性的情况下的可识别性感兴趣, 并且只观察到一个子节点。 它显示, 最高线型模型的参数是在一个标准下可识别的, 隐藏变量是容易和快速检查的。