Techniques for Discrimination-Free Predictive Models - Discrimination and Privacy in the Information Society - page 222

If the dataset D would have been unbiased; that is, S and Class were statistically
independent, the expected probability of being non-native and having the positive
class P ex p ( f ∧ +) would be:

= |

X ( S )= f |

× |

X ( Class )=+ |

P exp ( f ∧ +) : .

| D | | D |

For instance in the example dataset of Table 12.1, 50% of people are female, and
60% of people have a positive class. Therefore, if the dataset was non-discriminatory,
one would expect also 60% of females to have the positive class, which gives in total
50%

30% of people being female and having the positive class. In reality,
however, the observed probability in D ,

× 60% =

= |

X ( S )= f ∧ X ( Class )=+ |

P obs ( f ∧ +) :

| D |

might be different. If the expected probability is higher than the observed probabil-
ity value, it shows the bias towards class '

f .
Continuing the example, in the dataset of Table 12.1, we observe that only 2 people
in the dataset are female and have a positive class label, so the observed probability
of female and positive is 20%, which is considerably lower than the expected 30%,
thus indicating discrimination.
To compensate for the bias, we assign weights to objects. If a particular group
is under-represented, we give members of this group a higher weight, making them
more important in the classifier training process. The weight we assign to an object
is exactly the expected probability divided by the observed probability. In the ex-
ample this would mean that we assign a weight of 30% divided by 20% = 1.5 to
females with a positive class label. In this way we assign a weight to every object
according to its S -and Class -values. We call the dataset D with the added weights,
D W . It can be proven that the resulting dataset D W is unbiased; that is, if we multiply
the frequency of every object by its weight, the discrimination is 0. On this balanced
dataset the discrimination-free classifier is learnt.
Since not every classification algorithm can directly work with weights, we may
also use the weights when resampling the dataset; that is, we randomly select objects
from our training set to form a new dataset. When forming the new dataset, some
objects may be omitted and some may be duplicated. In the sampling procedure, the
weight of an object represents its relative chance of being chosen from the dataset;
that is, an object with a weight of 2.4 in every selection step has a 4 times higher
probability of being chosen than an object with a weight of 0.6. This variant is called
resampling .

− ' for those objects X with X ( S )=

Example 2. Consider again the dataset in Table 12.1. The weight for each data
object is computed according to its S- and Class-value, e.g. for instances with values
X ( Sex )= f and X ( Class )=+ :

0 . 5 × 0 . 6

W ( X )= = 1 . 5 .

0 . 2