The Annals of Statistics

Invariants of Some Probability Models Used in Phylogenetic Inference

Steven N. Evans and T. P. Speed

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The so-called method of invariants is a technique in the field of molecular evolution for inferring phylogenetic relations among a number of species on the basis of nucleotide sequence data. An invariant is a polynomial function of the probability distribution defined by a stochastic model for the observed nucleotide sequence. This function has the special property that it is identically zero for one possible phylogeny and typically nonzero for another possible phylogeny. Thus it is possible to discriminate statistically between two competing phylogenies using an estimate of the invariant. The advantage of this technique is that it enables such inferences to be made without the need for estimating nuisance parameters that are related to the specific mechanisms by which the molecular evolution occurs. For a wide class of models found in the literature, we present a simple algebraic formalism for recognising whether or not a function is an invariant and for generating all possible invariants. Our work is based on recognising an uderlying group structure and using discrete Fourier analysis.

Article information

Ann. Statist., Volume 21, Number 1 (1993), 355-377.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62H05: Characterization and structure theory
Secondary: 60K99: None of the above, but in this section 62F99: None of the above, but in this section

Invariant phylogenetic inference discrete Fourier analysis random walk on a group


Evans, Steven N.; Speed, T. P. Invariants of Some Probability Models Used in Phylogenetic Inference. Ann. Statist. 21 (1993), no. 1, 355--377. doi:10.1214/aos/1176349030.

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