1. Distance measures for skill vectors:

$d_{Euclidean} (s_1, s_2) ^2 = ||s_1 - s_2||^2 = 2(1- s_1\cdot s_2) = 2d_{cosine}(s_1, s_2) , \forall s_1, s_2 \in U^d$

2. Distances and Odds Ratios

Then the authors "show how a pairwise distance between two skill vectors is related to the logarithm of their odds ratio": $p = P( c = 1 | KS, s), o =\frac {p} {1-p}$ $\log (\frac {o_1}{o_2} ) ^2 = (KS \cdot \delta_{1, 2} )^2 \times 2dcosine(s_1, s_2)$ $\delta_{1, 2} = \frac {s_1 - s_2} {|| s_1 - s_2||}$