The Model-Specific Markov Embedding Problem for Symmetric Group-Based Models

Abstract

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.

The embedding problem for Markov matrices

“…A related problem is that of deciding when a transition matrix subjected to model constraints admits a Markov generator with the same restrictions on its entries. This problem has been deeply studied recently (see [Mat08,AKK21]).…”

“…The model embeddability for K3P matrices has has been largely discussed (and solved) in a more general context [AKK21] (see also [Mat08]). According to the previous lemma, the following characterizes K3P-embeddabable matrices in term of its eigenvalues.…”