No abstract

No abstract

We show that the expected size of the maximum agreement subtree of two n-leaf trees, uniformly random among all trees with the shape, is Θ( √ n). To derive the lower bound, we prove a global structural result on a decomposition of rooted binary trees into subgroups of leaves called blobs. To obtain the upper bound, we generalize a first moment argument from [1] for random tree distributions that are exchangeable and not necessarily sampling consistent.

Directed Gaussian graphical models are statistical models that use a directed acyclic graph (DAG) to represent the conditional independence structures between a set of jointly normal random variables. The DAG specifies the model through recursive factorization of the parametrization, via restricted conditional distributions. In this paper, we make an attempt to characterize the DAGs whose vanishing ideals are toric ideals. In particular, we give some combinatorial criteria to construct such DAGs from smaller DAGs which have toric vanishing ideals. An associated monomial map called the shortest trek map plays an important role in our description of toric Gaussian DAG models. For DAGs whose vanishing ideal is toric, we prove results about the generating sets of those toric ideals.

No abstract