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Lemma 3.1. Take a two-taxon discrete-time Markov model that is defined by a root distribution π m = ( π $^{m}$(1) . . . π $^{m}$(4) ) and two 4 × 4 probability transition matrices P ma and P $^{mb}$. Define Π m = diag π $^{m}$. Assume 

(a) P ma and P mb are strand-symmetric; 

(b) π m is compositionally asymmetric; and 

(c) P ma and P mb are invertible. 

Then π $^{m}$, P $^{ma}$, and P mb are uniquely determined by the joint probability matrix J ab = ( P mb ) $^{⊺}$Π $^{m}$Pma up to one of eight permutations of the states at node m . That is, the model is identifiable up to a suitable reordering of internal states. 

Example 3.1. It is interesting that a non-reversible, stationary, strand-symmetric model is not identifiable for two taxa, as can be shown by counterexample. Take such a model defined by 

<img src='content_image/313968.jpg'>

<img src='content_image/313969.jpg'>

and 

that yields the joint probability distribution J ab = ( P mb ) $^{⊺}$Π P $^{ma}$. The model defined 

