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Remark 3.2. The restriction that the mapping log P uv be unique may seem like a barrier to implementation of this model. It is difficult to imagine a sufficiently gen- eral parametrisation of Q uv that would guarantee this property. However, empirical evidence suggests that it may be better to beg forgiveness than to ask permission (Kaehler et al. 2015). Software exists for checking whether a 4 × 4 Markov gener- ator enjoys this property, and it was found that it was rarely necessary to actually enforce this constraint for the data sets considered in Kaehler et al. (2015). 

Proof. Again set 

<img src='content_image/318054.jpg'>

Note that if Q = SQS for a square matrix Q , then P = SPS for P = exp Q . This follows from the expansion P = ∑ ∞ i $_{=1}$( n !) − $^{1}$Q n and that Q n = SQ $^{n}$S because for example SQ $^{n}$S = SQSSQS ...SQS . The assumption that Q uv is strand-symmetric therefore also ensures that assumption D of Theorem 3.1 is satisfied. 

By Theorem 3.1, the mapping from the joint distribution of states at the terminal nodes of the tree to the root marginal probability distribution, the tree topology, and the transition probability matrices P uv is unique, so by assumption this property is also enjoyed by the transition rate matrices Q $^{uv}$. 

## 4 Reconstruction from Data: Consistency of Max- imum Likelihood 

As an application of the above identifiability results, we shall now prove that the above non-stationary, strand-symmetric models can be statistically consistently es- 

