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**Invariants of some probability models used in phylogenetic inference.**
*(English)*
Zbl 0772.92012

Summary: 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 underlying group structure and using discrete Fourier analysis.

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 underlying group structure and using discrete Fourier analysis.

### MSC:

92D15 | Problems related to evolution |

92D20 | Protein sequences, DNA sequences |

62P99 | Applications of statistics |

60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |

60G50 | Sums of independent random variables; random walks |

62H05 | Characterization and structure theory for multivariate probability distributions; copulas |

60K99 | Special processes |