ABSTRACT
In this section we present tree cumulants. The motivation is given by a simple example with three leaves, which we treat in detail below.
6.1.1 The tripod tree
Let T be the tripod tree in Figure 6.1 and let M(T , 2) be the corresponding general Markov model. Here the black vertices represent three observed variables X1, X2, X3, and the white vertex indicates a variable H that remains hidden. Since all variables in the system are binary, the parameterization is
p(x) =
θh(y)θ1|h(x1|y)θ2|h(x2|y)θ3|h(x3|y) for all x ∈ {0, 1}3. (6.1)
There are seven free parameters needed to specify p(x): θh(1) together with θ1|h(1|y), θ2|h(1|y) and θ3|h(1|y) for y = 0, 1. The Zariski closure of this model corresponds exactly to the first Secant Sec(1, 1, 1) as defined in Section 2.4; see also Example 2.60.
