Markovian log-supermodularity, and its applications in phylogenetics

TL;DR

This paper proves a log-supermodularity property for binary pattern distributions in Markov processes on trees and applies it to phylogenetics and combinatorial inequalities, enhancing understanding of evolutionary diversity and character analysis.

Contribution

It introduces a novel log-supermodularity property for Markov-generated distributions on trees and demonstrates its applications in phylogenetics and combinatorics.

Findings

Derived an inequality for expected phylogenetic diversity under extinction models

Established a combinatorial inequality for the parsimony score of binary characters

Proved the log-supermodularity property for distributions on tree tip patterns

Abstract

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.