.. -*- coding: utf-8 -*-

.. _bob.bio.demographics.fdr:

==========================
 Fairness Discrepancy Rate
==========================

Here we discuss the fairness discrepancy rate (FDR) proposed in::

   @article{de2020fairness,
      title={Fairness in Biometrics: a figure of merit to assess biometric verification systems},
      author={de Freitas Pereira, Tiago and Marcel, S{\'e}bastien},
      journal={arXiv preprint arXiv:2011.02395},
      year={2020}
   }



In this work, a biometric verification system is considered fair if statistical parity between groups is reached in terms of both `FMR` (False Match Rate) and
`FNMR` (False Non Match Rate) for a given decision threshold :math:`\tau`.
More formally, given a set of demographic groups :math:`\mathcal{D}=\{d_1,d_2,...,d_n\}`, and :math:`\tau = \text{FMR}_{x}`, a biometric verification system is considered
fair with respect to `FMR` if the following premisse holds:

.. math::

  \text{FMR}^{d_i}(\tau) \geq  \text{FMR}^{d_j}(\tau) - \epsilon \text{ } \forall d_i,d_j \in D


Such premisse can be written with the following equation:

.. math::

   A(\tau) = \max(|\text{FMR}^{d_i}(\tau)- \text{FMR}^{d_j}(\tau)|) \leq \epsilon \text{ } \forall d_i, d_j \in \mathcal{D}


Conversely, in terms of :math:`\text{FNMR}`, a biometric verification system is considered fair if the following premisse holds:

.. math::

     \text{FNMR}^{d_i}(\tau) \geq  \text{FNMR}^{d_j}(\tau) \text{ } \forall d_i,d_j \in D$.
   


Such premisse can be written with the following equation:


.. math::
   
    B(\tau) = \max(|\text{FNMR}^{d_i}(\tau)- \text{FNMR}^{d_j}(\tau)|) \leq \epsilon \text{ } \forall d_i, d_j \in \mathcal{D}.
   

Since `A` and `B` are functions of :math:`\tau`, both can be summarized in one figure of merit, that we refer as Fairness Discrepancy Rate (FDR) which is defined as:
   

.. math::

    FDR(\tau) = 1- (\alpha A(\tau) + (1-\alpha) B(\tau)),
    

where :math:`\alpha` is a hyper-parameter that defines the weight of :math:`A(\tau)` in the figure of merit (the importance of False Matches).
   


To see how FDR behaves in situations of fair/unfair score distributions, please check this `link <https://mybinder.org/v2/gh/tiagofrepereira2012/fdr/master?filepath=1.fair_unfair_example.ipynb>`_.