Maximum Likelihood

MLE Gaussian

class embedded_voting.RuleMLEGaussian(embeddings_from_ratings=None, tol=1e-06)

A rule that computes the scores of the candidates, assuming that the embeddings of the voters correspond to a covariance matrix.

For this rule, the embeddings must be a matrix n_voters * n_voters.

Examples

Consider a generating epistemic model, where the true value of each candidate is uniformly drawn in a given interval, and where the voters add a noise which is multivariate Gaussian.

>>> np.random.seed(42)
>>> covariance_matrix = np.array([
...     [2.02, 1.96, 0.86, 0.81, 1.67],
...     [1.96, 3.01, 1.46, 0.69, 1.59],
...     [0.86, 1.46, 0.94, 0.39, 0.7 ],
...     [0.81, 0.69, 0.39, 0.51, 0.9 ],
...     [1.67, 1.59, 0.7 , 0.9 , 1.78]
... ])
>>> ratings_generator = RatingsGeneratorEpistemicMultivariate(covariance_matrix=covariance_matrix)
>>> ratings = ratings_generator(n_candidates=2)
>>> ratings_generator.ground_truth_
array([17.73956049, 14.3887844 ])
>>> ratings
Ratings([[17.56232759, 14.51592899],
         [16.82544972, 15.78818081],
         [17.51952581, 14.44449175],
         [17.34964888, 14.4010885 ],
         [16.69480298, 14.9281998 ]])

If we know the covariance matrix of the noises, then RuleMLEGaussian is the maximum likelihood estimator of the ground truth:

>>> election = RuleMLEGaussian()(ratings, embeddings=covariance_matrix)
>>> election.scores_ # doctest: +ELLIPSIS
[268.6683142..., 221.5083075...]

Model Aware

class embedded_voting.RuleModelAware(groups_sizes, groups_features, group_noise=1, independent_noise=0)

A rule that is know the noise parameters of the model and use the maximum likelihood to select the best candidate.

Parameters:

• groups_sizes (list of int) – The number of voters in each group.
• groups_features (np.ndarray of shape (n_groups, n_features)) – The features of each group.
• group_noise (float) – The value of the feature noise.
• independent_noise (float) – The value of the distinct noise.

Examples

>>> ratings = Ratings(np.array([[.5, .6, .3], [.7, 0, .2], [.2, 1, .8]]))
>>> election = RuleModelAware([2, 1], [[1, 0], [0, 1]], group_noise=1, independent_noise=1)(ratings)
>>> election.ranking_
[1, 2, 0]
>>> election.scores_
[0.5, 0.7, 0.5666666...]
>>> election.winner_
1