Max-linear Bayesian networks have emerged as highly applicable models for causal inference from extreme value data. However, conditional independence (CI) for max-linear Bayesian networks behaves differently than for classical Gaussian Bayesian networks. We establish the parallel between the two theories via tropicalization, and establish the surprising result that the Markov equivalence classes for max-linear Bayesian networks coincide with the ones obtained by regular CI. Our paper opens up many open problems at the intersection of extreme value statistics, causal inference and tropical geometry.

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 $\Theta(\sqrt{n})$. 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 [D. I. Bernstein, et al., SIAM J. Discrete Math., 29 (2015), pp. 2065--2074] for random tree distributions that are exchangeable and not necessarily sampling consistent.

We prove lower bounds on the expected size of the maximum agreement subtree of two random binary phylogenetic trees under both the uniform distribution and the Yule--Harding distribution and prove upper bounds under the Yule--Harding distribution. This positively answers a question posed in earlier work. Determining tight upper and lower bounds remains an open problem.

Distance-based phylogenetic algorithms attempt to solve the NP-hard least-squares phylogeny problem by mapping an arbitrary dissimilarity map representing biological data to a tree metric. The set of all dissimilarity maps is a Euclidean space properly containing the space of all tree metrics as a polyhedral fan. Outputs of distance-based tree reconstruction algorithms such as UPGMA and neighbor-joining are points in the maximal cones in the fan. Tree metrics with polytomies lie at the intersections of maximal cones. A phylogenetic algorithm divides the space of all dissimilarity maps into regions based upon which combinatorial tree is reconstructed by the algorithm. Comparison of phylogenetic methods can be done by comparing the geometry of these regions. We use polyhedral geometry to compare the local nature of the subdivisions induced by least-squares phylogeny, UPGMA, and neighbor-joining when the true tree has a single polytomy with exactly four neighbors. Our results suggest that in some circumstances, UPGMA and neighbor-joining poorly match least-squares phylogeny.

We provide a logarithmic upper bound for the disentangling number on unordered lists of leaf labeled trees. This result is useful for analyzing phylogenetic mixture models. The proof depends on interpreting multisets of trees as high-dimensional contingency tables.

Phylogenetic data arising on two possibly different tree topologies might be mixed through several biological mechanisms, including incomplete lineage sorting or horizontal gene transfer in the case of different topologies, or simply different substitution processes on characters in the case of the same topology. Recent work on a 2-state symmetric model of character change showed that for 4 taxa, such a mixture model has nonidentifiable parameters, and thus, it is theoretically impossible to determine the two tree topologies from any amount of data under such circumstances. Here, the question of identifiability is investigated for two-tree mixtures of the 4-state group-based models, which are more relevant to DNA sequence data. Using algebraic techniques, we show that the tree parameters are identifiable for the JC and K2P models. We also prove that generic substitution parameters for the JC mixture models are identifiable, and for the K2P and K3P models obtain generic identifiability results for mixtures on the same tree. This indicates that the full phylogenetic signal remains in such mixtures, and the 2-state symmetric result is thus a misleading guide to the behavior of other models.

We construct examples of contingency tables on n binary random variables where the gap between the linear programming lower/upper bound and the true integer lower/upper bounds on cell entries is exponentially large. These examples provide evidence that linear programming may not be an effective heuristic for detecting disclosures when releasing margins of multiway tables.