Kybernetika 56 no. 6, 1154-1175, 2020

Brownian motion tree models are toric

Bernd Sturmfels, Caroline Uhler and Piotr ZwiernikDOI: 10.14736/kyb-2020-6-1154


Felsenstein's classical model for Gaussian distributions on a phylogenetic tree is shown to be a toric variety in the space of concentration matrices. We present an exact semialgebraic characterization of this model, and we demonstrate how the toric structure leads to exact methods for maximum likelihood estimation. Our results also give new insights into the geometry of ultrametric matrices.


Brownian motion tree model, ultrametric matrices, toric geometry


62R01, 62H22, 15B48


  1. T. W. Anderson: Estimation of covariance matrices which are linear combinations or whose inverses are linear combinations of given matrices. In: {Essays in {P}robability and {S}tatistics} (I.|,M. Mahalanobis, P. C. Rao, C. R. Bose, R. C. Chakravarti and K. J. C. Smith, eds.), Univ. of North Carolina Press, Chapel Hill, 1970, pp. 1-24.   CrossRef
  2. L. Bossinger, X. Fang, G. Fourier, M. Hering and M. Lanini: Toric degenerations of Gr(2,n) and Gr(3,6) via plabic graphs. Ann. Combinator. 22 (2018), 3, 491-512.   DOI:10.1007/s00026-018-0395-z
  3. P. Buneman: The recovery of trees from measures of dissimilarity. In: Mathematics in the Archaeological and Historical Sciences (F. Hodson et al., ed.), Edinburgh University Press, 1971, pp. 387-395.   CrossRef
  4. D. Carlson and T. L. Markham: Schur complements of diagonally dominant matrices. Czechosl. Math. J. 29 (1979), 2, 246-251.   CrossRef
  5. C. Dellacherie, S. Martinez and J. San Martin: Inverse M-matrices and ultrametric matrices. Springer 2118, 2014.   DOI:10.1007/978-3-319-10298-6\_1
  6. J. Draisma and J. Kuttler: On the ideals of equivariant tree models. Math. Ann. 344 (2009), 3, 619-644.   DOI:10.1007/s00208-008-0320-6
  7. J. S. Sullivant and K. Talaska: Positivity for {G}aussian graphical models. Adv. Appl. Math. 50 (2013), 5, 661-674.   DOI:10.1016/j.aam.2013.03.001
  8. J. Felsenstein: Maximum-likelihood estimation of evolutionary trees from continuous characters. Amer. J. Human Genetics 25 (1973), 5, 471-492.   CrossRef
  9. D. Grayson and M. Stillman: Macaulay2, a software system for research in algebraic geometry.    CrossRef
  10. K. Kaveh and Ch. Manon: Khovanskii bases, higher rank valuations and tropical geometry. SIAM J. Appl. Algebra Geometry 3 (2019), 2, 292-336.   DOI:10.1016/j.aam.2013.03.001
  11. D. Maclagan and B. Sturmfels: Introduction to Tropical Geometry. American Mathematical Society, Graduate Studies in Mathematics 161, Providence 2015.   DOI:10.1090/gsm/161
  12. M. Michałek, B. Sturmfels, C. Uhler and P. Zwiernik: Exponential varieties. Proc. London Math. Soc. (3), 112 (2016), 1, 27-56.   DOI:10.1112/plms/pdv066
  13. Ch. Semple and M. Steel: Phylogenetics. Oxford University Press, 2003.   DOI:10.1080/10635150490888895
  14. V. Moulton and M. Steel: Peeling phylogenetic 'oranges'. Adv. App. Mathemat. 33 (2004), 4, 710-727.   DOI:10.1016/j.aam.2004.03.003
  15. S. Sullivant, K. Talaska and J. Draisma: Trek separation for {G}aussian graphical models. Ann. Stat. 38 (2010), 3, 1665-1685.   DOI:10.1214/09-aos760
  16. R. S. Varga and R. Nabben: On symmetric ultrametric matrices. Numerical Linear Algebra (L. Reichel et al., eds.), de Gruyter, New York 1993, pp. 193-199.   DOI:10.1515/9783110857658.193
  17. P. Zwiernik, C. Uhler and D. Richards: Maximum likelihood estimation for linear Gaussian covariance models. J. Roy. Stat. Soc.: Series B (Stat. Method.) 79 (2017), 4, 1269-1292.   DOI:10.1111/rssb.12217