.. _about_fairness:

About Algorithmic Fairness
==========================

Below, see the mathematical definition for each of the fairness metrics in the library.

Average Odds
^^^^^^^^^^^^
Average Odds denotes the average of difference in FPR and TPR for group 1 and group 2.

.. math::
    \frac{1}{2} [(FPR_{D = \text{group 1}} - FPR_{D =
    \text{group 2}}) + (TPR_{D = \text{group 2}} - TPR_{D
    = \text{group 1}}))]

Disparate Impact
^^^^^^^^^^^^^^^^
Disparate Impact is the ratio of predictions for a "positive" outcome in a binary classification task between members of group 1 and group 2, respectively.

.. math::

    \frac{P(\hat{Y} = 1 | D = \text{group 1})}
        {P(\hat{Y} = 1 | D = \text{group 2})}

Equal Opportunity
^^^^^^^^^^^^^^^^^
Equal Opportunity calculates the ratio of true positives to positive examples in the dataset, :math:`TPR = TP/P`, conditioned on a protected attribute.

FNR Difference
^^^^^^^^^^^^^^
FNR Difference measures the equality (or lack thereof) of the false negative rates across groups. In practice, this metric is implemented as a difference between the metric value for group 1 and group 2.

.. math::

    E[d(X)=0 \mid Y=1, g(X)] = E[d(X)=0, Y=1]

FOR Difference
^^^^^^^^^^^^^^
FOR Difference measures the equality (or lack thereof) across groups of the rate of inaccurate "negative" predictions by the model. It is calculated using the ratio of false negatives to negative examples in the dataset, :math:`FOR = FN/N`, conditioned on a protected attribute.

Generalized Entropy Index
^^^^^^^^^^^^^^^^^^^^^^^^^
Generalized Entropy Index is proposed as a unified individual and group fairness measure in [1]_. With :math:`b_i = \hat{y}_i - y_i + 1`:

.. math::

           \mathcal{E}(\alpha) = \begin{cases}
              \frac{1}{n \alpha (\alpha-1)}\sum_{i=1}^n\left[\left(\frac{b_i}{\mu}\right)^\alpha - 1\right] &
              \alpha \ne 0, 1, \\
              \frac{1}{n}\sum_{i=1}^n\frac{b_{i}}{\mu}\ln\frac{b_{i}}{\mu} & \alpha=1, \\
            -\frac{1}{n}\sum_{i=1}^n\ln\frac{b_{i}}{\mu},& \alpha=0.
            \end{cases}

References:
            .. [1] T. Speicher, H. Heidari, N. Grgic-Hlaca, K. P. Gummadi, A. Singla, A. Weller, and M. B. Zafar,
             A Unified Approach to Quantifying Algorithmic Unfairness: Measuring Individual and Group Unfairness via
             Inequality Indices, ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2018.

Predictive Equality
^^^^^^^^^^^^^^^^^^^
Predictive Equality is defined as the situation when accuracy of decisions is equal across two groups, as measured by false positive rate (FPR).

.. math::

    E[d(X)|Y=0, g(X)] = E[d(X), Y=0]

Statistical Parity
^^^^^^^^^^^^^^^^^^
Statistical Parity measures the difference in probabilities of a positive outcome across two groups.
 
.. math::

    P(\hat{Y} = 1 | D = \text{group 1}) - P(\hat{Y} = 1 | D = \text{group 2})

Theil Index
^^^^^^^^^^^
Theil Index is the generalized entropy index with :math:`\alpha = 1`. See Generalized Entropy Index.


Equalized Odds
^^^^^^^^^^^^^^

Equalized odds is a bias mitigation technique where subset of decisions of a binary classifier is flipped at uniform random in each of two groups to achieve equality of TPR and FPR across the two groups as proposed in [2]_. This subset rate in each group is learned via constrained optimization.

References:
            .. [2] Moritz Hardt, Eric Price, and Nathan Srebro. 2016. Equality of opportunity in supervised learning. In Proceedings of the 30th International Conference on Neural Information Processing Systems (NIPS'16).