TY - JOUR
T1 - Phase transition in the sample complexity of likelihood-based phylogeny inference
AU - Roch, Sebastien
AU - Sly, Allan
N1 - Funding Information:
Allan Sly: Work partly done at Microsoft Research. Work supported by NSF Grants DMS-1208339 and DMS-1352013 and an Alfred P. Sloan Research Fellowship.
Funding Information:
Sebastien Roch: Work partly done at Microsoft Research, UCLA, IPAM and the Simons Institute for the Theory of Computing. Work supported by NSF Grants DMS-1007144 and DMS-1149312 (CAREER), and an Alfred P. Sloan Research Fellowship.
Publisher Copyright:
© 2017, Springer-Verlag GmbH Germany.
PY - 2017/10/1
Y1 - 2017/10/1
N2 - Reconstructing evolutionary trees from molecular sequence data is a fundamental problem in computational biology. Stochastic models of sequence evolution are closely related to spin systems that have been extensively studied in statistical physics and that connection has led to important insights on the theoretical properties of phylogenetic reconstruction algorithms as well as the development of new inference methods. Here, we study maximum likelihood, a classical statistical technique which is perhaps the most widely used in phylogenetic practice because of its superior empirical accuracy. At the theoretical level, except for its consistency, that is, the guarantee of eventual correct reconstruction as the size of the input data grows, much remains to be understood about the statistical properties of maximum likelihood in this context. In particular, the best bounds on the sample complexity or sequence-length requirement of maximum likelihood, that is, the amount of data required for correct reconstruction, are exponential in the number, n, of tips—far from known lower bounds based on information-theoretic arguments. Here we close the gap by proving a new upper bound on the sequence-length requirement of maximum likelihood that matches up to constants the known lower bound for some standard models of evolution. More specifically, for the r-state symmetric model of sequence evolution on a binary phylogeny with bounded edge lengths, we show that the sequence-length requirement behaves logarithmically in n when the expected amount of mutation per edge is below what is known as the Kesten-Stigum threshold. In general, the sequence-length requirement is polynomial in n. Our results imply moreover that the maximum likelihood estimator can be computed efficiently on randomly generated data provided sequences are as above. Our main technical contribution, which may be of independent interest, relates the total variation distance between the leaf state distributions of two trees with a notion of combinatorial distance between the trees. In words we show in a precise quantitative manner that the more different two evolutionary trees are, the easier it is to distinguish their output.
AB - Reconstructing evolutionary trees from molecular sequence data is a fundamental problem in computational biology. Stochastic models of sequence evolution are closely related to spin systems that have been extensively studied in statistical physics and that connection has led to important insights on the theoretical properties of phylogenetic reconstruction algorithms as well as the development of new inference methods. Here, we study maximum likelihood, a classical statistical technique which is perhaps the most widely used in phylogenetic practice because of its superior empirical accuracy. At the theoretical level, except for its consistency, that is, the guarantee of eventual correct reconstruction as the size of the input data grows, much remains to be understood about the statistical properties of maximum likelihood in this context. In particular, the best bounds on the sample complexity or sequence-length requirement of maximum likelihood, that is, the amount of data required for correct reconstruction, are exponential in the number, n, of tips—far from known lower bounds based on information-theoretic arguments. Here we close the gap by proving a new upper bound on the sequence-length requirement of maximum likelihood that matches up to constants the known lower bound for some standard models of evolution. More specifically, for the r-state symmetric model of sequence evolution on a binary phylogeny with bounded edge lengths, we show that the sequence-length requirement behaves logarithmically in n when the expected amount of mutation per edge is below what is known as the Kesten-Stigum threshold. In general, the sequence-length requirement is polynomial in n. Our results imply moreover that the maximum likelihood estimator can be computed efficiently on randomly generated data provided sequences are as above. Our main technical contribution, which may be of independent interest, relates the total variation distance between the leaf state distributions of two trees with a notion of combinatorial distance between the trees. In words we show in a precise quantitative manner that the more different two evolutionary trees are, the easier it is to distinguish their output.
KW - Maximum likelihood
KW - Phylogenetic reconstruction
KW - Sequence-length requirement
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U2 - 10.1007/s00440-017-0793-x
DO - 10.1007/s00440-017-0793-x
M3 - Article
AN - SCOPUS:85026804156
SN - 0178-8051
VL - 169
SP - 3
EP - 62
JO - Probability Theory and Related Fields
JF - Probability Theory and Related Fields
IS - 1-2
ER -