Variance on the Leaves of a Tree Markov Random Field: Detecting Character Dependencies in Phylogenies

TL;DR

This paper introduces methods to detect dependencies between characters in phylogenetic trees modeled by Markov random fields, extending traditional models that assume independence, and provides new variance bounds for character count variables.

Contribution

It develops techniques to identify significant correlations between characters in phylogenies under certain assumptions, including a new variance concentration result for multistate variables.

Findings

Detects significant, history-independent correlations between characters.

Proves a new variance bound for character count variables on trees.

Extends analysis to models with dependent characters, especially CFN model.

Abstract

Stochastic models of evolution (Markov random fields on trivalent trees) generally assume that different characters (different runs of the stochastic process) are independent and identically distributed. In this paper we take the first steps towards addressing dependent characters. Specifically we show that, under certain technical assumptions regarding the evolution of individual characters, we can detect any significant, history independent, correlation between any pair of multistate characters. For the special case of the Cavender-Farris-Neyman (CFN) model on two states with symmetric transition matrices, our analysis needs milder assumptions. To perform the analysis, we need to prove a new concentration result for multistate random variables of a Markov random field on arbitrary trivalent trees: we show that the random variable counting the number of leaves in any particular subset of states has variance that is subquadratic in the number of leaves.