BradleyTerry

crowdkit.aggregation.pairwise.bradley_terry.BradleyTerry | Source code

BradleyTerry(
    self,
    n_iter: int,
    tol: float = 1e-05
)

Bradley-Terry, the classic algorithm for aggregating pairwise comparisons.

This algorithm constructs an items' ranking based on pairwise comparisons. Given a pair of two items $i$ and $j$, the probability of $i$ to be ranked higher is, according to the Bradley-Terry's probabilitstic model,

$$P(i > j) = \frac{p_i}{p_i + p_j}.$$

Here $\boldsymbol{p}$ is a vector of positive real-valued parameters that the algorithm optimizes. These optimization process maximizes the log-likelihood of observed comparisons outcomes by the MM-algorithm:

$$L(\boldsymbol{p}) = \sum_{i=1}^n\sum_{j=1}^n[w_{ij}\ln p_i - w_{ij}\ln (p_i + p_j)],$$

where $w_{ij}$ denotes the number of comparisons of $i$ and $j$ "won" by $i$.

David R. Hunter. MM algorithms for generalized Bradley-Terry models Ann. Statist., Vol. 32, 1 (2004): 384–406.

Bradley, R. A. and Terry, M. E. Rank analysis of incomplete block designs. I. The method of paired comparisons. Biometrika, Vol. 39 (1952): 324–345.

Parameters Description

Examples:

The Bradley-Terry model needs the data to be a DataFrame containing columns left, right, and label. left and right contain identifiers of left and right items respectfuly, label contains identifiers of items that won these comparisons.

import pandas as pd
from crowdkit.aggregation import BradleyTerry
df = pd.DataFrame(
    [
        ['item1', 'item2', 'item1'],
        ['item2', 'item3', 'item2']
    ],
    columns=['left', 'right', 'label']
)

Methods Summary