holisticai.bias.metrics.exposure_kl

holisticai.bias.metrics.exposure_kl(group_a, group_b, mat_pred, top=None, thresh=0.5, normalize=False)

Exposure KL Divergence

This function computes the KL divergence from the group_a exposure distribution to the group_b exposure distribution.

Interpretation

A KL divergence of 0 is desired, which occurs when the distributions are equal. Higher values of the KL divergence indicate difference in exposure distributions of group a and group b.

Parameters

group_aarray-like

Group membership vector.

group_barray-like

Group membership vector.

mat_predmatrix-like

Matrix with shape (num_users, num_items). A recommender score (binary or soft pred) for each user,item interaction.

topint, optional

If not None, the number of items that are shown to each user.

threshfloat, optional

Threshold indicating value at which a given item is shown to user (if top is None).

normalizebool, optional

If True, normalises the data matrix to [0,1] range.

Returns

float

Exposure KL Divergence

References

[1]

A Dash, A Chakraborty, S Ghosh, A Mukherjee, K P. Gummadi (2021). When the Umpire is also a Player: Bias in Private Label Product Recommendations on E-commerce Marketplaces, <https://arxiv.org/pdf/2102.00141.pdf >

[2]

T Giannakas, P Sermpezis, A Giovanidis, T Spyropoulos, G Arvanitakis (2021). Fairness in Network-Friendly Recommendations, <https://arxiv.org/pdf/2102.00141.pdf >

Examples

>>> import numpy as np
>>> from holisticai.bias.metrics import exposure_kl
>>> group_a = np.array([1, 1, 1, 1, 0, 0, 0, 0, 0, 0])
>>> group_b = np.array([0, 0, 0, 0, 1, 1, 1, 1, 1, 1])
>>> mat_pred = np.array([[0.9, 0.8, 0.4, 0.2],
                        [0.7, 0.9, 0.1, 0.7],
                        [0.3, 0.2, 0.3, 0.3],
                        [0.2, 0.1, 0.7, 0.8],
                        [0.8, 0.7, 0.9, 0.1],
                        [1. , 0.9, 0.3, 0.6],
                        [0.8, 0.9, 0.1, 0.1],
                        [0.2, 0.3, 0.1, 0.5],
                        [0.1, 0.2, 0.7, 0.7],
                        [0.2, 0.7, 0.1, 0.2]])
>>> exposure_kl(group_a, group_b, mat_pred, top=1, thresh=None, normalize=False)
0.23217831296817806