Information geometry for phylogenetic trees

M. K. Garba, T. M. W. Nye, J. Lueg, S. F. Huckemann*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)
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Abstract

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.
Original languageEnglish
Article number19
Number of pages39
JournalJournal of Mathematical Biology
Volume82
Issue number3
Early online date15 Feb 2021
DOIs
Publication statusPublished - Feb 2021
Externally publishedYes

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