The optimal rate for resolving a nearpolytomy in a phylogeny

Steel, Mike
Biomathematics Research Centre, University of Canterbury, Christchurch, New Zealand

Leuenberger, Christoph
Département de mathématiques, Université de Fribourg, Switzerland
Published in:
 Journal of Theoretical Biology.  2017, vol. 420, p. 174–179
English
The reconstruction of phylogenetic trees from discrete character data typically relies on models that assume the characters evolve under a continuoustime Markov process operating at some overall rate λ. When λ is too high or too low, it becomes difficult to distinguish a short interior edge from a polytomy (the tree that results from collapsing the edge). In this note, we investigate the rate that maximizes the expected loglikelihood ratio (i.e. the Kullback–Leibler separation) between the fourleaf unresolved (star) tree and a fourleaf binary tree with interior edge length ϵ. For a simple twostate model, we show that as ϵ converges to 0 the optimal rate also converges to zero when the four pendant edges have equal length. However, when the four pendant branches have unequal length, two local optima can arise, and it is possible for the globally optimal rate to converge to a nonzero constant as ϵ→0ϵ→0. Moreover, in the setting where the four pendant branches have equal lengths and either (i) we replace the twostate model by an infinitestate model or (ii) we retain the twostate model and replace the Kullback–Leibler separation by Euclidean distance as the maximization goal, then the optimal rate also converges to a nonzero constant.

Faculty
 Faculté des sciences et de médecine

Department
 Département de Mathématiques

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Mathematics

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https://folia.unifr.ch/unifr/documents/305579
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