The model-specific Markov embedding problem for symmetric group-based models

We study model embeddability, which is a variation of the famous embedding problem in probability theory, when apart from the requirement that the Markov matrix is the matrix exponential of a rate matrix, we additionally ask that the rate matrix follows the model structure. We provide a characterisa...

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Bibliographic Details
Published in:Journal of mathematical biology 2021-09, Vol.83 (3), p.33-33, Article 33
Main Authors: Ardiyansyah, Muhammad, Kosta, Dimitra, Kubjas, Kaie
Format: Article
Language:English
Subjects:
Algebra
Applications of Mathematics
Biology
Deoxyribonucleic acid
DNA
Eigenvalues
Embedding
Evolution
Markov analysis
Markov Chains
Mathematical analysis
Mathematical and Computational Biology
Mathematics
Mathematics and Statistics
Matrices (mathematics)
Phylogenetics
Phylogeny
Probability theory
Random variables
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Summary:We study model embeddability, which is a variation of the famous embedding problem in probability theory, when apart from the requirement that the Markov matrix is the matrix exponential of a rate matrix, we additionally ask that the rate matrix follows the model structure. We provide a characterisation of model embeddable Markov matrices corresponding to symmetric group-based phylogenetic models. In particular, we provide necessary and sufficient conditions in terms of the eigenvalues of symmetric group-based matrices. To showcase our main result on model embeddability, we provide an application to hachimoji models, which are eight-state models for synthetic DNA. Moreover, our main result on model embeddability enables us to compute the volume of the set of model embeddable Markov matrices relative to the volume of other relevant sets of Markov matrices within the model.
ISSN:0303-6812
1432-1416
DOI:10.1007/s00285-021-01656-5