Bradley-Terry-Luce (BTL)

The Bradley-Terry-Luce (BTL) model is well-known in the literature on discrete choice (Bradley & Terry, 1952). It starts from the parametric model of pairwise comparisons:

$\mathbb{P}( \boldsymbol{x}_i \succ \boldsymbol{x}_j ) = \dfrac{ \theta_i }{ \theta_i + \theta_j } \, .$

Given n x n matrix C of pairwise comparisons, where C[i,j] informs about how often the item x_i wins over x_j, the parameters

$\large \boldsymbol{\theta} = ( \theta_1, \theta_2, \dots, \theta_n ) \in \mathbb{R}^n_+ \, ,$

can be estimated by likelihood maximization:

$\large \hat{ \boldsymbol{\theta} } \in \underset{{ \boldsymbol{\theta} \in \mathbb{R}^{n} }}{\arg \max} \prod_{1 \leq i \neq j \leq n} \left( \dfrac{\theta_{i}}{\theta_{i} + \theta_{j}} \right)^{c_{i,j}} \, ,$

the implementation of which is placed in BTL.py. Check out this work for more details.

mahmadif/rank-aggregation

Bradley-Terry-Luce (BTL)