Bradley-Terry Model

Bradley-Terry model fitting via maximum likelihood.

The BT model assumes P(i beats j) = sigma(theta_i - theta_j) where sigma is the logistic function. Fitting recovers the strength vector theta.

class winference.bradley_terry.BradleyTerry(models)

Bases: object

Maximum-likelihood Bradley-Terry model.

Parameters:

models (list[str]) – Unique model identifiers.

Examples

>>> bt = BradleyTerry(["A", "B", "C"])
>>> bt.fit(comparisons)  # list of (i, j, outcome) tuples
>>> bt.win_probability("A", "B")
0.73

fit(comparisons, reg=0.0001)

Fit the model from a list of (model_a, model_b, a_wins) triples.

Parameters:

• comparisons (list[tuple[str, str, bool]]) – Each element is (model_a, model_b, a_wins).

• reg (float, default: 0.0001) – L2 regularisation strength (prevents divergence for unbeaten models).

Return type:

Self

Returns:

Self for method chaining.

win_probability(model_a, model_b)

Predicted P(model_a beats model_b).

Parameters:

• model_a (str)

• model_b (str)

Return type:

float

win_probability_matrix()

NxN matrix of predicted win probabilities.

Return type:

ndarray[tuple[Any, ...], dtype[double]]

predicted_win_rates(comparisons)

Return predicted P(a wins) for each comparison triple.

Parameters:

comparisons (list[tuple[str, str, bool]])

Return type:

ndarray[tuple[Any, ...], dtype[double]]

strengths()

Return {model: theta} dictionary.

Return type:

dict[str, float]

rank()

Models sorted by decreasing strength.

Return type:

list[str]

winference.bradley_terry.fit_bt_from_matrix(win_matrix, count_matrix, models, reg=0.0001)

Convenience: fit BT from win/count matrices rather than triples.

Parameters:

• win_matrix (ndarray[tuple[Any, ...], dtype[double]])

• count_matrix (ndarray[tuple[Any, ...], dtype[double]])

• models (list[str])

• reg (float, default: 0.0001)

Return type:

BradleyTerry