论文标题
代数两级测量树
Algebraic two-level measure trees
论文作者
论文摘要
借助代数树，Löhr和Winter（2021）引入了图理论树的概念的概括，以解释潜在的不可数的结构。树结构由地图给出，将其分配给其分支点的每个三倍。不考虑边缘长度或距离。可以在Borel-$σ$ field上配备树木的自然拓扑和概率度量，并以这种方式定义一个代数测量树。 Löhr和Winter的主要结果是提供样品形状收敛在二元代数测量树空间上的紧凑拓扑。这是通过用圆的三角形编码后者来证明的。在本文中，我们将此结果扩展到两个级别的设置。通过研究具有两个生物学水平的层次系统（例如宿主 - 寄生虫种群）的动机，我们为代数树配备了一系列概率指标的概率度量。为了显示二进制代数两级测量树的空间的紧凑性，我们通过在圆线上添加两级度量来丰富这些树的编码。作为一种应用，我们定义了两级代数的金曼树，这是从嵌套的金曼合并获得的随机代数两级测量树。
With the algebraic trees, Löhr and Winter (2021) introduced a generalization of the notion of graph-theoretic trees to account for potentially uncountable structures. The tree structure is given by the map which assigns to each triple of points their branch point. No edge length or distance is considered. One can equip a tree with a natural topology and a probability measure on the Borel-$σ$-field, defining in this way an algebraic measure tree. The main result of Löhr and Winter is to provide with the sample shape convergence a compact topology on the space of binary algebraic measure trees. This was proved by encoding the latter with triangulations of the circle. In the present paper, we extend this result to a two level setup. Motivated by the study of hierarchical systems with two levels in biology, such as host-parasite populations, we equip algebraic trees with a probability measure on the set of probability measures. To show the compactness of the space of binary algebraic two-level measure trees, we enrich the encoding of these trees by triangulations of the circle, by adding a two-level measure on the circle line. As an application, we define the two-level algebraic Kingman tree, that is the random algebraic two-level measure tree obtained from the nested Kingman coalescent.
