Edge log-ratios in a Bradley-Terry comparison tree (${\eta }_{ij}=\log ({\theta }_i/{\theta }_j)$)

Orient the edges of a comparison tree and set ${\eta }_{ij}=\log {\theta }_i-\log {\theta }_j$. After anchoring one vertex strength, these edge quantities are unconstrained coordinates. The Bradley-Terry model probability is the logistic function of ${\eta }_{ij}$, and its binomial log-likelihood contribution is $m[p{\eta }_{ij}-\log (1+e^{{\eta }_{ij}})]$. Differentiating gives the fitted odds $p/(1-p)$.