Markovian log-supermodularity, and its applications in phylogenetics

We establish a log-supermodularity property for probability distributions on binary patterns observed at the tips of a tree that are generated under any 2-state Markov process. We illustrate the applicability of this result in phylogenetics by deriving an inequality relevant to estimating expected future phylogenetic diversity under a model of species extinction. In a further application of the log-supermodularity property, we derive a purely combinatorial inequality for the parsimony score of a binary character. The proofs of our results exploit two classical theorems in the combinatorics of finite sets.