Maximum Likelihood Estimation for Brownian Motion Tree Models Based on One Sample

We study the problem of maximum likelihood estimation given one data sample (n “ 1) over Brownian Motion Tree Models (BMTMs), a class of Gaussian models on trees. BMTMs are often used as a null model in phylogenetics, where the one-sample regime is common. Specifically, we show that, almost surely, the one-sample BMTM maximum likelihood estimator (MLE) exists, is unique, and corresponds to a fully observed tree. Moreover, we provide a polynomial time algorithm for its exact computation. We also consider the MLE over all possible BMTM tree structures in the one-sample case and show that it exists almost surely, that it coincides with the MLE over diagonally dominant M-matrices, and that it admits a unique closed-form solution that corresponds to a path graph. Finally, we explore statistical properties of the one-sample BMTM MLE through numerical experiments.

[1]  Sébastien Roch,et al.  A short proof that phylogenetic tree reconstruction by maximum likelihood is hard , 2005, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[2]  J. Felsenstein.  Phylogenies and the Comparative Method , 1985, The American Naturalist.

[3]  Louis J. Billera,et al.  Geometry of the Space of Phylogenetic Trees , 2001, Adv. Appl. Math..

[4]  Olivier Ledoit,et al.  A well-conditioned estimator for large-dimensional covariance matrices , 2004 .

[5]  Yulia Mostovoy,et al.  Inferring Evolutionary Histories of Pathway Regulation from Transcriptional Profiling Data , 2013, PLoS Comput. Biol..

[6]  Shripad Tuljapurkar,et al.  Limitations of GCTA as a solution to the missing heritability problem , 2015, Proceedings of the National Academy of Sciences.

[7]  Paul Barford,et al.  Network radar: tomography from round trip time measurements , 2004, IMC '04.

[8]  C. Dellacherie,et al.  Inverse M-Matrices and Ultrametric Matrices , 2014 .

[9]  Vincent Y. F. Tan,et al.  Learning Latent Tree Graphical Models , 2010, J. Mach. Learn. Res..

[10]  N. Saitou,et al.  The neighbor-joining method: a new method for reconstructing phylogenetic trees. , 1987, Molecular biology and evolution.

[11]  Daniel Pérez Palomar,et al.  Minimax Estimation of Laplacian Constrained Precision Matrices , 2021, AISTATS.

[12]  Robert P Freckleton,et al.  Detecting Non-Brownian Trait Evolution in Adaptive Radiations , 2006, PLoS biology.

[13]  L. Brown.  Fundamentals of statistical exponential families: with applications in statistical decision theory , 1986 .

[14]  S. Lauritzen,et al.  Maximum likelihood estimation in Gaussian models under total positivity , 2017, The Annals of Statistics.

[15]  Bernd Sturmfels,et al.  Brownian motion tree models are toric , 2019, Kybernetika.

[16]  Art M. Duval,et al.  Simplicial matrix-tree theorems , 2008, 0802.2576.

[17]  J. Felsenstein,et al.  EVOLUTIONARY TREES FROM GENE FREQUENCIES AND QUANTITATIVE CHARACTERS: FINDING MAXIMUM LIKELIHOOD ESTIMATES , 1981, Evolution; international journal of organic evolution.

[18]  S. Bergmann,et al.  The evolution of gene expression levels in mammalian organs , 2011, Nature.

[19]  Jiaxi Ying,et al.  Does the $\ell_1$-norm Learn a Sparse Graph under Laplacian Constrained Graphical Models? , 2020, 2006.14925.

[20]  Caroline Uhler,et al.  Maximum likelihood estimation for linear Gaussian covariance models , 2014, 1408.5604.

[21]  D. Dey,et al.  Estimation of a covariance matrix under Stein's loss , 1985 .

[22]  J. Felsenstein.  Maximum-likelihood estimation of evolutionary trees from continuous characters. , 1973, American journal of human genetics.

[23]  M. Okamoto.  Distinctness of the Eigenvalues of a Quadratic form in a Multivariate Sample , 1973 .

[24]  Olivier Ledoit,et al.  Nonlinear Shrinkage Estimation of Large-Dimensional Covariance Matrices , 2011, 1207.5322.

[25]  N. Cooper,et al.  Body Size Evolution in Mammals: Complexity in Tempo and Mode , 2010, The American Naturalist.

[26]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[27]  Paul Barford,et al.  Toward the Practical Use of Network Tomography for Internet Topology Discovery , 2010, 2010 Proceedings IEEE INFOCOM.