Introducing Positive Discrimination in Predictive Models - Discrimination and Privacy in the Information Society - page 255

Intuitively, this approach makes a lot of sense since it uses different classifiers
to classify data that is known to be differently distributed (males are different from
females). Since males are still favored, however, the resulting classification can
still contain discrimination. We apply the threshold modification algorithm to re-
move this discrimination.

14.4.3 A Latent Variable Model

Our third and most sophisticated approach tries to model the discrimination
process in order to discover the actual class labels that the data-set should have
contained if it would have been discrimination-free. Since they are not observed,
these actual class labels are modeled using a latent (or hidden) variable, see, e.g.,
(Bishop, 2006). Such a latent variable can be seen as an attribute that is known to
exist, but its values have not been recorded in the data-set. A well-known example
of such a variable is “happiness”. It is very difficult to observe if someone is hap-
py, but since we known how being happy influences one's actions, we can infer
whether someone is happy by observing his or her actions. In our case, we cannot
know who should have gotten a positive class label, but we can make assumptions
about how this variable depends on the other variables:
1. The actual discrimination-free class label is independent from the sensitive
attribute.
2. The observed class label is determined by discriminating the actual labels based
on the sensitive attribute uniformly at random.

These two assumptions might not correspond to how discrimination is being applied
in practice. For instance, the females close to the decision boundary could have a
higher chance of being discriminated. However, because they result in a simple
model, they do allow us to study the problem of discrimination-free classification in
detail. The resulting model is given by the following total probability function:

P(C,L,S,A 1 ,A 2 ,…,A n ) = P(L)P(S)P(C|L,S)P(A 1 |L,S)P(A 2 |L,S)…P(A n |L,S),
where C is the class attribute after discrimination, L is the latent variable
representing the true class before discrimination, S is the sensitive attribute, and
A 1 , A 2 ,…, A n are all other attributes. The formula is similar to the original Naive
Bayes formula in the sense that all attributes A 1 , A 2 ,…, A n are independent from
each other given the class label. Except that in this model, we use the actual latent
class label L instead of C. In addition, every value except L is conditioned on the
sensitive attribute S. The result is identical to the previous approach that used two
separate models; for every value of S, a different set of probability functions are
used, thus a different model is used for every value of S. The distribution of L
however, is modeled to be independent from S, satisfying the first assumption.
The probability function P(C|L,S) satisfies the second assumption: for every com-
bination of an actual latent class label value with a sensitive value, a different
probability function is used to determine the observed class label. Thus, the
uses classifiers to classify data from different distributions. Also, in our experience it
produces worse results.