ABSTRACT
In this chapter we obtain the full description of these models in terms of implicit polynomial equations and inequalities. Like in the previous chapter, the focus is on the two-state case M(T , 2). In this case an important part of the information about the global geometry of the general Markov model is contained in its second-order moments. By Theorem 6.10, the correlations of p ∈M(T , 2) satisfy
ρij = ∏ e∈ij
ρe for all 1 ≤ i < j ≤ m. (7.1)
Here we used the fact that second-order standardized tree cumulants coincide with correlations; see Remark 6.3. From this we easily infer the following set of constraints that must be satisfied by any distribution in M(T , 2). Lemma 7.1. Let T be a phylogenetic tree with labeling set [m]. For every p ∈M(T , 2) the corresponding correlations satisfy
ρijρikρjk ≥ 0 for all 1 ≤ i < j < k ≤ m, or equivalently
kijkikkjk ≥ 0 for all 1 ≤ i < j < k ≤ m. Later in this chapter we supplement these inequalities with the complete
set of constraints describing M(T , 2) for any phylogenetic tree T .
