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Information geometry for phylogenetic trees

Lookup NU author(s): Maryam Garba, Dr Tom Nye



This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).


© 2021, The Author(s).We propose a new space of phylogenetic trees which we call wald space. The motivation is to develop a space suitable for statistical analysis of phylogenies, but with a geometry based on more biologically principled assumptions than existing spaces: in wald space, trees are close if they induce similar distributions on genetic sequence data. As a point set, wald space contains the previously developed Billera–Holmes–Vogtmann (BHV) tree space; it also contains disconnected forests, like the edge-product (EP) space but without certain singularities of the EP space. We investigate two related geometries on wald space. The first is the geometry of the Fisher information metric of character distributions induced by the two-state symmetric Markov substitution process on each tree. Infinitesimally, the metric is proportional to the Kullback–Leibler divergence, or equivalently, as we show, to any f-divergence. The second geometry is obtained analogously but using a related continuous-valued Gaussian process on each tree, and it can be viewed as the trace metric of the affine-invariant metric for covariance matrices. We derive a gradient descent algorithm to project from the ambient space of covariance matrices to wald space. For both geometries we derive computational methods to compute geodesics in polynomial time and show numerically that the two information geometries (discrete and continuous) are very similar. In particular, geodesics are approximated extrinsically. Comparison with the BHV geometry shows that our canonical and biologically motivated space is substantially different.

Publication metadata

Author(s): Garba MK, Nye TMW, Lueg J, Huckemann SF

Publication type: Article

Publication status: Published

Journal: Journal of Mathematical Biology

Year: 2021

Volume: 82

Issue: 3

Print publication date: 01/02/2021

Online publication date: 15/02/2021

Acceptance date: 21/10/2020

Date deposited: 14/12/2020

ISSN (print): 0303-6812

ISSN (electronic): 1432-1416

Publisher: Springer Science and Business Media Deutschland GmbH


DOI: 10.1007/s00285-021-01553-x


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