Section 4 of the KQN paper derives a probabilistic skill similarity measure. (1) Distance measures for skill vectors: because skill vectors lie in $U^d$, the squared Euclidean distance between two skill vectors equals twice their cosine distance: $d_{Euclidean}(s_1, s_2)^2 = ||s_1 - s_2||^2 = 2(1 - s_1 \cdot s_2) = 2 d_{cosine}(s_1, s_2), ; \forall s_1, s_2 \in U^d$ (2) Distances and odds ratios: the authors show that the pairwise distance between two skill vectors relates to the logarithm of their odds ratio. With correctness probability $p = P(c = 1 \mid KS, s)$ and odds $o = \frac{p}{1-p}$: $\left(\log \frac{o_1}{o_2}\right)^2 = (KS \cdot \delta_{1,2})^2 \times 2 d_{cosine}(s_1, s_2), \quad \delta_{1,2} = \frac{s_1 - s_2}{||s_1 - s_2||}$