Most Parsimonious Likelihood Exhibits Multiple Optima for Compatible Characters


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.


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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