Most Parsimonious Likelihood Exhibits Multiple Optima for Compatible Characters

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Maximum likelihood estimators are a popular method for scoring phylogenetic trees to best explain the evolutionary histories of biomolecular sequences. In 1994, Steel showed that, given an incompatible set of binary characters and a fixed tree topology, there exist multiple sets of branch lengths that are optima of the maximum average likelihood estimator. Since parsimony techniques—another popular method of scoring evolutionary trees—tend to exhibit favorable behavior on data compatible with the tree, Steel asked if the same is true for likelihood estimators, or if multiple optima can occur for compatible sequences. We show that, despite exhibiting behavior similar to parsimony, multiple local optima can occur for compatible characters for the most parsimonious likelihood estimator. We caution that thorough understanding of likelihood criteria is necessary before they are used to analyze biological data.

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  1. 1.

    Our goal with the use of the term “sequence” is to call back to the scientific process of obtaining observations in a continuous setting. We expect likelihood estimates to change as more samples are discovered and characterized. We do not use the term to refer to unaligned DNA sequences for a single taxon.

  2. 2.

    Note that \(I_{{\mathrm{MP}}}\) yields the first lump given in Table 1.


  1. Barry D, Hartigan J (1987) Statistical analysis of hominoid molecular evolution. Stat Sci 2:191–207

  2. Bininda-Emonds ORP, Gittleman JL, Steel MA (2002) The (super) tree of life. Ann Rev Ecol Syst 33:265–89

  3. Charleston MA, Perkins SL (2003) Lizards, malaria, and jungles in the caribbean. In: Page RD (ed) Tangled trees: phylogeny, cospeciation, and coevolution. University of Chicago Press, Chicago, pp 65–92

  4. Chor B, Hendy MD, Holland BR, Penny D (2000) Multiple maxima of likelihood in phylogenetic trees: an analytic approach. Mol Biol Evol 17(10):1529–1541

  5. Foulds LR, Graham RL (1982) The Steiner problem in phylogeny is NP-complete. Adv Appl Math 3(1):43–49

  6. Foundation PS (2010) Python language reference, version 2.7. Accessed 29 Apr 2019

  7. Fukami K, Tateno Y (1989) On the maximum likelihood method for estimating molecular trees: uniqueness of the likelihood point. J Mol Evol 28(5):460–464

  8. Gusfield D (1991) Efficient algorithms for inferring evolutionary trees. Networks 21(1):19–28

  9. Hillis DM, Mable BK, Moritz C (1996) Molecular systematics. Sinauer Associates, Sunderland

  10. Janies DA, Treseder T, Alexandrov B, Habibb F, Chen JJ, Ferreira R, Catalyurek U, Varon A, Wheeler WC (2011) The supramap project: linking pathogen genomes with geography to fight emergent infectious diseases. Cladistics 27:61–66

  11. Roch S (2006) A short proof that phylogenetic tree reconstruction by maximum likelihood is hard. IEEE/ACM Trans Comput Biol Bioinform 3(1):92–94

  12. Semple C, Steel M (2003) Phylogenetics. Oxford lecture series in mathematics and its applications, vol 24. Oxford University Press, Oxford

  13. Steel M (2011) The penny ante challenge problems: open problems from the New Zealand phylogenetics meetings. Accessed 8 Aug 2019

  14. Steel MA (1994) The maximum likelihood point for a phylogenetic tree is not unique. Syst Biol 43(4):560–564

  15. Steel M, Penny D (2000) Parsimony, likelihood, and the role of models in molecular phylogenetics. Mol Biol Evol 17(6):839–850

  16. Stein W et al (2015) Sage mathematics software (version 6.6). The Sage Development Team. Accessed 8 Aug 2019

  17. Stewart J (2005) Multivariable calculus: concepts and contexts. Brooks/Cole, Pacific Grove ISBN 0-534-41004-9

  18. Tuffley C, Steel M (1997) Links between maximum likelihood and maximum parsimony under a simple model of site substitution. Bull Math Biol 59(3):581–607

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We would like to thank Dan Gusfield, Rob Gysel, Mike Steel, and Ward Wheeler for helpful comments and conversations. This work was partially supported by a grant from the Simons Foundation to KS.

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Correspondence to Julia Matsieva.

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Matsieva, J., St. John, K. Most Parsimonious Likelihood Exhibits Multiple Optima for Compatible Characters. Bull Math Biol 82, 10 (2020).

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  • Models of evolution
  • Phylogenetic trees
  • Maximum likelihood estimators
  • Maximum parsimony criteria