Score-Based Density Estimation from Pairwise Comparisons

We study density estimation from pairwise comparisons, motivated by expert knowledge elicitation and learning from human feedback. We relate the unobserved target density to a tempered winner density (marginal density of preferred choices), learning the winner's score via score-matching. This allows estimating the target by `de-tempering' the estimated winner density's score. We prove that the score vectors of the belief and the winner density are collinear, linked by a position-dependent tempering field. We give analytical formulas for this field and propose an estimator for it under the Bradley-Terry model. Using a diffusion model trained on tempered samples generated via score-scaled annealed Langevin dynamics, we can learn complex multivariate belief densities of simulated experts, from only hundreds to thousands of pairwise comparisons.