Additionally assume that for every { u, v } ∈ E , where v is further from the root than u , that there exists a unique mapping Q uv = log P $^{uv}$, where every off-diagonal element of Q uv is non-negative. Also replace assumption D from Theorem 3.1 with the assumption that Q uv = SQ $^{uv}$S . 

Then the method of maximum likelihood consistently recovers the topology, root marginal probability distribution, and edge transition rate matrices. That is, for any θ ∈ Θ , let ˆ θ$_{n}$ denote the maximum likelihood estimate based on n independent observations { X i $_{T}$}$_{i}$$_{∈{}$$_{1}$$_{,...,n}$$_{}}$ of character states at the terminal nodes of the tree, then ˆ θ$_{n}$ a.s. −→ θ as n → ∞ . 

The following is a restatement of Lemma 5.1 in Chang (1996), where a sketch of its proof is provided. 

Lemma 4.1. Let X be a finite set and let {P$_{θ}$}$_{θ}$$_{∈}$$_{Θ}$ be a set of probability distributions on X , where the closure Θ of Θ is a compact subset of a metric space. Let { X$_{i}$ }$_{i}$$_{∈}$ Z be independent and identically distributed random variables (or vectors) with probability distribution P$_{θ}$$_{0}$ for some θ$_{0}$ ∈ Θ . Assume the identifiability condition 

P$_{θ}$ ̸ = P$_{θ}$$_{0}$ for each θ ∈ Θ with θ ̸ = θ$_{0}$. 

Suppose that for each x ∈ X the function θ ↦→ P$_{θ}$ ( x ) is continuous on Θ and let ˆ θ$_{n}$ = ˆ θ$_{n}$ ( X$_{1}$, . . . , X$_{n}$ ) maximise the log likelihood ∑ n i $_{=1}$log P$_{θ}$ ( X$_{i}$ ) over θ ∈ Θ . Then under P$_{θ}$$_{0}$ , ˆ θ$_{n}$ a.s. −→ θ$_{0}$ as n → ∞ . 

Proof of Theorem 4.1. The proof of Theorem 5.1 in Chang (1996) carries to this context with only slight modification. We need to show that the conditions of Lemma 4.1 are implied by the assumptions of Theorem 4.1 with X = { A, C, G, T } | T | and P$_{θ}$ ( A ) = P$_{θ}$ ( X$_{T}$ ∈ A ) for A ⊆ X . 

For θ, θ$_{0}$ ∈ Θ, Theorem 3.1 gives that P$_{θ}$ ̸ = P$_{θ}$$_{0}$ if θ ̸ = θ$_{0}$ . It remains to show for θ$_{0}$ ∈ Θ and θ ∈ Θ that P$_{θ}$ ̸ = P$_{θ}$$_{0}$ if θ ̸ = θ$_{0}$ . 

