Abstract
We establish necessary and sufficient conditions for consistent root reconstruction in continuous-time Markov models with countable state space on bounded-height trees. Here a root state estimator is said to be consistent if the probability that it returns to the true root state converges to $1$ as the number of leaves tends to infinity. We also derive quantitative bounds on the error of reconstruction. Our results answer a question of Gascuel and Steel [GS10] and have implications for ancestral sequence reconstruction in a classical evolutionary model of nucleotide insertion and deletion [TKF91].
Highlights
In biology, the inferred evolutionary history of organisms and their relationships is depicted diagrammatically as a phylogenetic tree, that is, a rooted tree whose leaves represent living species and branchings indicate past speciation events [Fel04]
We are concerned with the problem of inferring an ancestral state from observations at the leaves of a given tree under known Markovian dynamics
Main results Here, we study the root reconstruction problem in an alternative setting where estimators with stronger properties can be derived
Summary
Background In biology, the inferred evolutionary history of organisms and their relationships is depicted diagrammatically as a phylogenetic tree, that is, a rooted tree whose leaves represent living species and branchings indicate past speciation events [Fel04]. We consider sequences of nested trees with uniformly bounded depths This is motivated by contemporary applications in evolutionary biology where the rapidly increasing availability of data from ever-growing numbers of organisms, genome sequencing data, has allowed dense sampling of species within the same family or genus. We give both necessary and sufficient conditions for consistent root reconstruction for general trees and general Markov processes on countable state spaces, a question left open in [GS10]. Our main theorem immediately gives necessary and sufficient conditions for the existence of consistent root estimators for a classical such model known as the TKF91 process [TKF91].
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