Definition. Bradley--Terry comparison law [bradley1952rank, Page 325, equation (1)] [ftip-002Q]
Definition. Bradley--Terry comparison law [bradley1952rank, Page 325, equation (1)] [ftip-002Q]
Given scalar scores for two alternatives, the Bradley--Terry comparison law assigns the probability
\[ \Pr _\phi (y^+\succ y^-\mid x) =\frac {\exp r_\phi (x,y^+)} {\exp r_\phi (x,y^+)+\exp r_\phi (x,y^-)} =\sigma \left (r_\phi (x,y^+)-r_\phi (x,y^-)\right ), \]where \(\sigma (a)=(1+\exp (-a))^{-1}\). Conditioning the scores on a language-model prompt is the contextual specialization used in modern reward modeling; see also [rafailov2023direct, Section 3, equation (1)].