In our experiments with the Adult Census dataset, found out that on a GPU with
$4Go$ of memory, we can fit up to $3$ different running instances of $FM{}$
with some residual memory left, $4$ instances of $GANSan{}$, $GANSan{}-OM$ and
$WGAN{}$, and $2$ instances of $AttGAN{}$. As a consequence, the
hyperparameters exploration can be a bit faster for $FM{}$.

##  7\. Conclusion

In this paper, we proposed a novel approach Fair Mapping that transforms a
given data point onto a chosen target distribution while preventing the
possibility of inferring the sensitive attribute. In particular, our approach
preserves the realistic aspect of the transformed dataset while providing the
transformed data points with the same benefit as the privileged group from the
point of view of a trained classifier. As a consequence, a classifier already
trained does not need to be re-trained to use the transformed dataset, which
is usually required with other pre-processing approaches. Nevertheless, our
approach is not suitable for training a fair classifier that will be used on
the unmodified original data. Indeed, as our procedure transport all
distribution unto the target one, the resulting dataset is only composed of a
single distribution. A classifier trained on the transformed dataset would not
be suitable for deploiement where the data are used without first being
processed trough our approach, since the classifier has not been trained on
other distributions than the target one.

Our experimental results shows that our approach is able to protect the
sensitive attribute and outperforms some approach of the state of the art. The
mutual information further enforce the sensitive attribute protection,
especially in case of more than two sensitive groups. We have also observed,
by comparison with the state of the art, that the protection of the sensitive
attribute can be better carried out when some degree of freedom is given to a
protected distribution, thus increasing the space of possible representations.
In addition, our approach also improves some other fairness metrics used in
the literature, as we have demonstrated in our experiments.

Nonetheless, some improvement can be carried out in our approach, especially
on the generalisation and the stability of our framework during the cross-
validation. A larger exploration of the hyperparameters space would further
enhance the quality of our results.

A future direction of research would consist in the verification of such
property, as well as the ability to work on groups that were not seen during
the training of models (_i.e._ , Black-Female), but can be obtained by
combining known sensitive values (_i.e._ , Black-Male and White-Female).

##  8\. Relationship between the group Fairness and prevention of inference of
the sensitive attribute

In this section, we discuss the relationship between the group fairness and
the prevention of inference of the sensitive attribute. For the demonstration,
let’s assume that $A,Y$ are independent to $S$. Thus
$P(A=a,Y=y,S=s)=P(A=a,Y=y)P(S=s)$. Furthermore, let’s assume that we have
access to a predictor $f$, that infer $y$ from $A$ : $\hat{Y}=f(A)$. Since
$A,Y$ and $S$ are independent, so will be $f(A)$, since $f$ only uses $A,Y$.
The demographic parity states:

(24) $\begin{cases}DP&=P(\hat{Y}=y/S=s_{i})-P(\hat{Y}=y/S=s_{j})<\epsilon\\\
&=P(f(A)=y/S=s_{i})-P(f(A)=y/S=s_{j})<\epsilon\\\
&\forall\>\\{s_{i},s_{j}\\}\in S\\\
EO&=P(\hat{Y}=y/S=s_{i},Y=y)-P(\hat{Y}=y/S=s_{j},Y=y)<\epsilon\\\
&=P(f(A)=y/S=s_{i},Y=y)-P(f(A)=y/S=s_{j},Y=y)<\epsilon\\\
&\forall\>\\{s_{i},s_{j}\\}\in S\end{cases}$

with $y=1$.

The Independence assumption, allow us to write $P(f(A)=y/S=s_{i})=P(f(A)=y)$
and $P(f(A)=y/Y=y,S=s_{i})=P(f(A)=y)/Y=y$. By plug-in it in equation 8 we
obtain

$\begin{cases}DP&=P(f(A)=y/S=s_{i})-P(f(A)=y/S=s_{j})\\\
&=P(f(A)=y)-P(f(A)=y)=0\\\ EO&=P(f(A)=y/S=s_{i},Y=y)-P(f(A)=y/S=s_{j},Y=y)\\\
&=P(f(A)=y)/Y=y-P(f(A)=y)/Y=y)=0\end{cases}$

Thus, if $A,Y$ and $S$ are independent, metrics such as the accuracy and the
equalized odds would be satisfied. Fairness preprocessing approaches that
works by preventing the inference of the sensitive attribute aim to achieve
the independence between $S$ and $A$ (most of these approaches assume that $Y$
is independent to $S$) by removing the correlations between the sensitive
attribute and the other variable. As a consequence, they should also improve
the demographic parity and the equalized odds.

##  9\. Transformation of the data decision attribute

Throughout this paper, we only considered the case where the decision
attribute is not modified with our approach. Nonetheless, it is important to
remark that our approach can also transform the decision attribute, further
enhancing the overall fairness as the approach would find the most appropriate
decision for each transformed individual on the privileged distribution. As a
matter of fact, if the protected and the privileged distribution significantly
differs in terms of decision rates due to any form of discrimination or
favouritism, the approach would apply the same treatment to all datapoints.
Thus, leading to the correct decision had the individuals belonging to the
privileged group on which data are mapped onto. Unfortunately, by transforming
the decision attribute, we lose the groundtruth for the decision prediction,
as the modified decision is an altered one, that might not correspond to any
realistic record.

##  10\. Optimal BER value for preventing the sensitive attribute inference

Suppose that a dataset is composed of $k$ different sensitive groups, with
$P(S=s_{x})$ the probability or proportion of each group, and $\hat{S}$ the
predicted sensitive attribute. The best protection is achieved when the
classifier behaves as a random guess of each class. As such, we can consider
the classifier as a random sampling of the sensitive attribute with
replacement. Let’s consider that the proportion of each class $s_{x}$ obtained
through the prediction or sampling of the random classifier predict is
$P(\hat{S}=s_{x})$. As the probability of sampling successfully a member of
class $s_{x}$ is driven with $P(S=s_{x})$, for each class
$s_{x}\>;\>x\in\\{1,\ldots,k\\}$, the law of large number state that
$P(\hat{S}=s_{x})$ will become closer to $P(S=s_{x})$. Meaning that in the
long run, the random classifier will predict each class $s_{x}$ with
probability $P(S=s_{x})$.

As we assumed the classifier behaving as a random guessing, the prediction of
$S$ is independent of the real value $S$. No information on $\hat{S}$ can be
obtained by observing $S$. Thus, $P(\hat{S}=s_{x}\mid
S=s_{x})=P(\hat{S}=s_{x})=P(S=s_{x})$ as previously shown.

As a consequence, $BER{}=1-\dfrac{1}{k}\left(\sum_{i=1}^{k}P(\hat{S}=s_{x}\mid
S=s_{x})\right)$ (cf. equation 3), becomes
$BER{}=1-\dfrac{1}{k}\left(\sum_{i=1}^{k}P(S=s_{x})\right)=1-\dfrac{1}{k}\left(1\right)=1-\dfrac{1}{k}=\dfrac{k-1}{k}$

Maximizing the protection with the $BER{}$ thus correspond to minimizing
$\dfrac{k-1}{k}-BER{}=\dfrac{-1}{k}+\dfrac{1}{k}\left(\sum_{i=1}^{k}P(\hat{S}=s_{x}\mid
S=s_{x})\right)$.

##  11\. Mutual information computation

Given the sensitive attribute $S$ with values
$s_{i}\>;\>i\in\\{1,\ldots,k\\}$, and its prediction $\hat{S}$ made by a
classifier $f$, the computation of the mutual information between $S$ and
$\hat{S}$ requires access to the joint distribution $P(S,\hat{S})$, and the
marginals $P(S)$ and $P(\hat{S})$. Since we are dealing with the discrete set
$S$, the joint distribution can be computed by counting the number of elements
in each set ($S=s_{i}\cap\hat{S}=s_{j}$, $i,\>j\>\in\\{1,\ldots,k\\}$),
divided by the number of total elements $\mid S\mid=k$ :
$P(S=s_{i}\cap\hat{S}=s_{j})=\dfrac{\mid S=s_{i}\cap\hat{S}=s_{j}\mid}{k}$.
Similarly, the joint probability can be obtained through
$P(S=s_{i}\cap\hat{S}=s_{j})=P(\hat{S}=s_{j}\mid S=s_{i})*P(S=s_{i})$

Assume that the predictor $f$ outputs the probability of each datapoint to
belong to each group $s_{i}$. Thus, for each individual $t$ (in a dataset $R$
of $N$ individuals) with data $r_{t}$, $f$ outputs $\langle
p_{t}^{1},\ldots,p_{t}^{k}\rangle$, with $p_{t}^{j}=f(r_{t})^{j}$ the
probability that $r_{t}$ has the sensitive value of $s_{j}$. The group
predicted by $f$ can be obtained by selecting the group with the highest
probability ($argmax$). However, the $argmax$ function is not differentiable.

The joint distribution for FM is computed by using the probabilities obtained
through $f$ ($p_{t}^{i}$), instead of counting the exact group obtained
through $f$. $P(S=s_{i}\cap\hat{S}=s_{j})$ is obtained with equation 25.

(25) $\begin{split}P(S=s_{i}\cap\hat{S}=s_{j})&=P(\hat{S}=s_{j}\mid
S=s_{i})*P(S=s_{i})\\\ &=\dfrac{\sum_{r_{t}\in R/S=s_{i}}f(r_{t})^{j}}{\mid
r_{t}\in R/s_{i}\mid}*P(S=s_{i})\end{split}$

, with $r_{t}\in R/S=s_{i}$ the set of datapoints in the dataset with the
sensitive attribute $s_{i}$, $f(r_{t})^{j}$ the probability that $r_{t}$ has
the sensitive attribute $s_{j}$ as given with $f$, and $P(S=s_{i})$ the
proportion of the data with sensitive value $s_{i}$. The marginal
$P(\hat{S}=s_{j})$ can be obtained with a similar procedure, by averaging over
the predicted probabilities of $f$ (equation 26).

(26)
$\begin{split}P(\hat{S}=s_{j})=\dfrac{\sum_{t=1}^{N}f(r_{t})^{j}}{N}\end{split}$

##  12\. Hyperparameters for external classifiers

In Table 8 we present the external classifiers and their respective
hyperparameters.

Table 8. Hyperparameters used for all external classifiers. Parameters not
mentioned are used with default values

Classifier | Parameters
---|---
Gradient Boosting Classifier (GBC) | Estimators | $100$
Learning rate | $0.1$
Maximum depth | $3$
Multi-layer Perceptron Classifier (MLPC) | Default parameters
hidden layer sizes | $100$
activation | $relu$
solver | $adam$
Decision Tree (DTC) | Default parameters
criterion | $gini$
splitter | $best$
max depth | $None$

##  13\. Experiment results with original privileged data

In this section, we present the results of all approaches investigated in this
paper when using the original data from the privileged group. These results
are, for most approaches, obtained by replacing the reconstructed values of
the privileged group by their original counterpart. We refer the reader to the
Table 2 for the optimal protection metrics values.

###  13.1. Lipton results

Figure 15. Pareto fronts in the Fair Mapping perspective for approaches
investigated on Lipton. The right column present all points on the fronts,
while the right columns present the same results but on the range [$0.99$ \-
$1$], for a better visualization. Top to bottom: protection
$BER{}_{og_{prv}}$, protection $SAcc_{og_{prv}}$, accuracy of $S$,
$Pc_{prot}$, mutual information $MI_{og_{prv}}$.

On the Lipton dataset (Figure 15), we can observe that all approaches exhibit
a similar behaviour as observed with the reconstructed privileged data.
However, the protection achieved by all approaches are lower than the one
achieved with the reconstructed privileged group. This observation is
particularly true for $GANSan{}-OM$, where the model displays the highest
protection among all approaches, even though the fidelity of the privileged
group is lower. WGAN transforms the data such that it resemble the privileged
group, but still fail to protect the sensitive attribute.

###  13.2. German Credit results

Figure 16. Pareto fronts in the Fair Mapping perspective for approaches
investigated on German. The right column present all points on the fronts,
while the right columns present the same results but on the range [$0.94$ \-
$1$], for a better visualisation.

On German credit (Figure 16), the use of the original privileged data only
slightly improves the protection of the sensitive attribute. Only $FM{}$ and
$GANSan{}-OM$ slightly improves the baseline result. All other approaches do
not. the transformation is largely dominated by $AttGAN{}$, closely followed
by $FM{}$. Unfortunately, $DIRM{}-OM$ neither improves the protection nor the
transformation ($Pc_{prot}$).

###  13.3. Adult Census results

Figure 17. Pareto fronts in the Fair Mapping perspective for approaches
investigated on Adult census. The right column present all points on the
fronts, while the right columns present the same results but on the range
[$0.97$ \- $1$], for a better visualisation.

On Adult Census (Figure 17), the protection of the sensitive attribute with
the original data from the privileged group is nearly impossible for all
approaches. No approaches successfully provide a protection better than the
baseline as measured with either the $BER{}$ or the $SAcc{}$, even though some
approaches such as $WGAN{}$ or $FM{}$ can successfully maximize the
transformation metric $Pc_{prot}$. This suggests that a minimum level of
modification in the privileged group is necessary to protect the sensitive
attribute.

###  13.4. Adult2 Census with results

Figure 18. Pareto fronts in the Fair Mapping perspective for approaches
investigated on Adult census with 2 attributes. The right column present all
points on the fronts, while the right columns present the same results but on
the range [$0.99$ \- $1$], for a better visualization.

Just as observed with Adult on a single binary attribute, we can observe
(Figure 18) that the protection of two attributes without the modification of
the privileged group is highly difficult. Even though we observe that we can
achieve a higher BER value, the accuracy is still higher than the baseline
computed with the original data. The mutual information is still significantly
higher than measured on the original data. No approaches provide an
appropriate level of protection. This observation suggest that in case of
multiple attribute, the $BER{}$ can not be the used as the only protection
measure, and certain amount of modification of the privileged data is required
in order to improve the data protection.

###  13.5. FM instantiations

We present in Figure 19 the different instances of FM. We can observe that all
of those approaches have an almost similar range of values. The figure also
demonstrates the stability of our approach.

Figure 19. Different Fair Mapping instances. We can observe that the different
$FM{}$ instances behave almost similarly for all chosen loss function for
protecting the sensitive attribute. Each column represent the performances on
a single dataset. On Adult2, we always include the mutual information
regularisation.

##  14\. Divergences with original data

In Figure 20 we present the Sinkhorn divergences results with respect to the
protection computed with the original privileged group $BER{}_{og_{prv}}$,
while Figure 21 present the same results with respect to the fidelity of the
privileged data.

Figure 20. Divergences computed on German, Adult, Adult2 and Lipton. Each
column represent a dataset while each row represent the divergence between the
transformed protected data and respectively the original protected data
($R_{prot}\sim G_{w}(R_{prot})$), the reconstructed privileged group
($G_{w}(R_{priv})\sim G_{w}(R_{prot})$), and the original privileged group
($R_{priv}\sim G_{w}(R_{prot})$). The divergences are represented with respect
to the protection $BER{}_{og_{prv}}$.  Figure 21. Divergences computed on
German, Adult, Adult2 and Lipton. Each column represent a dataset while each
row represent the divergence between the transformed protected data and
respectively the original protected data ($R_{prot}\sim G_{w}(R_{prot})$), the
reconstructed privileged group ($G_{w}(R_{priv})\sim G_{w}(R_{prot})$), and
the original privileged group ($R_{priv}\sim G_{w}(R_{prot})$). The
divergences are represented with respect to the fidelity $Fid{}_{priv}$.

##  15\. AutoEncoder fidelity and protection

In this section, we discuss the difference of results between the protection
obtained with the original privileged data (presented in section 6) and the
reconstructed privileged data (section 13). From those results, we can observe
that a minimum amount of modification of the privileged data is required in
order to enhance the sensitive attribute protection, while any method working
the original privileged data cannot improve the results significantly above
the baseline. In order to evaluate whether such important difference can be
explained with other factors than our training procedure, we trained an
AutoEncoder to reconstruct the privileged group data. For different fidelity
values, we trained a classifier to distinguish the reconstructed privileged
group from their original values. The underlying idea is that if the trained
classifier are always able to distinguish the reconstructed data from their
original version, the AutoEncoder introduce some form of watermark within the
reconstructed data that render it always distinguishable from their original
values. As such, we cannot expect the transforming model to be able to
transport the protected distribution unto the privileged one without such
watermark. The ideal values to achieve are an accuracy of $0.5$ and a $BER{}$
of $0.5$ for fidelity values different from $1$.

Figure 22. Accuracy, BER and MI of classifiers trained to differentiate the
original privileged group from their reconstructed version with a vanilla (no
other constraint except the reconstruction) AutoEncoder.

The results are presented in Figure 22. We can observe that for dataset Lipton
and German, unless the fidelity obtained is well above $0.999489$ and $0.9907$
respectively, the trained classifier have a $BER{}$ lower than $0.4$ and an
accuracy greater than $0.64$ and $0.77$ respectively. In our results,
achieving such fidelity values is highly difficult for any of the chosen
approaches. On Lipton, the accuracy and the $BER{}$ are well below $0.66$ and
above $0.36$ respectively, for any fidelity values. This suggest that our
transformation mechanism could transform the protected group into the
privileged data such that protection measured reach similar values, which,
unfortunately, is not exactly the case. On Adult on the other hand, the lowest
accuracy is $0.7559$ and the highest $BER$ is $0.2440$. The lowest fidelity is
$0.9979$ and the highest is $0.9981$. In our results, the highest fidelity
with our approach are $0.9967$ and $0.9988$ on Adult2 and Adult. As a
consequence, we cannot expect the transformation to produce data
indistinguishable from the original privileged distribution.

##  16\. Results on other approaches perspective

In this section, we will observe the performances of different approaches
through different perspectives.

###  16.1. Lipton results

Figure 23. Results on the perspective of WGAN (lipton dataset), which is the
transformation of data such that they resemble the privileged group.

In the WGAN perspective (Figure 23) which is the transformation of data from
the protected group, we can observe that nearly all approaches that have the
privileged group as a chosen target distribution achieve a transformation
performance above or nearly equal to $0.8$. The exception is with $DIRM{}-OM$.
As mentioned in section 6, this result might be a consequence of the
underlying repair scheme (procedure) or procedure of the $DIRM{}$ approach.

Figure 24. Results on the perspective of AttGAN, (Lipton dataset)which is the
transformation of data such that they resemble the privileged group.

On the AttGAN perspective (Figure 24), as we expect from $GANSan{}$ and
$DIRM{}$ the transformation results are the lowest. $AttGAN{}$ dominates for
higher fidelities values, but are equal to $FM{}$ and $WGAN{}$ for lower
values.

Figure 25. Results on the perspective of GANSan and DIRM (lipton dataset),
which is the transformation of data such that they resemble the privileged
group.

GANSan and DIRM shares the same perspective (Figure 25). We can observe that
GANSan provides the best trade-offs between protection and data
reconstruction. $FM{}$ has a lower fidelity as we could expect, since there
are no reconstruction constraint on the protected group, which we only want to
render similar to the privileged group distribution. Nonetheless, it is
interesting to observe that by only constraining the model to reconstruct the
privileged group, it has enough information to significantly reduce the error
in the reconstruction of the protected data.

###  16.2. German Credit results

Figure 26. Results on the perspective of WGAN (German dataset), which is the
transformation of data such that they resemble the privileged group.

Just as observed with the Lipton dataset, the best results in the WGAN
perspective (Figure 26) are achieved with $WGAN{}$, $FM{}$ and $AttGAN{}$. All
other approaches have performances close to $0.7$. Surprisingly, $GANSan{}$
outperforms $DIRM{}-OM$ and $GANSan{}-OM$ on the transformation metric.

Figure 27. Results on the perspective of AttGAN (German dataset), which is the
transformation of data such that they resemble the privileged group.

On the AttGAN perspective (Figure 27) AttGAN and FM are the most dominant
approaches.

Figure 28. Results on the perspective of GANSan (German dataset), which is the
transformation of data such that they resemble the privileged group.

FM dominates all approaches on the GANSan and DIRM perspective. GANSan
slightly improves the fidelity with a little enhancement of the protection,
while DIRM produces results close to the baseline. Even if the protection is
slightly enhanced with GANSan, we can observe that the mutual information is
still high. A notable observation is that GANSan and FM completes each other
in this perspective. In fact, where FM fall short in terms of fidelity on all
the dataset datapoints, GANSan slightly improves such fidelity. Both Pareto-
fronts nearly forms a single continuous one.

###  16.3. Adult Census results

Figure 29. Results on the perspective of WGAN (Adult dataset), which is the
transformation of data such that they resemble the privileged group.

On the WGAN perspective (Figure 29), as observed on any other dataset, $FM{}$
and $WGAN{}$ are maximizing the transformation metric. Surprisingly, AttGAN
does not have a performance better than $90\%$, even though it also has the
objective of maximizing the transformation metric.

Figure 30. Results on the perspective of AttGAN (Adult dataset), which is the
transformation of data such that they resemble the privileged group.

In the perspective of AttGAN in Figure 30, FM and AttGAN behave almost in a
complementary fashion. AttGAN slightly dominates FM on higher fidelity values.
Surprisingly, DIRM and DIRM-OM dominates GANSan-OM.

Figure 31. Results on the perspective of GANSan (Adult dataset), which is the
transformation of data such that they resemble the privileged group.

On the GANSan perspective (Figure 31), it is interesting to see that FM and
AttGAN still behave in a complementary fashion. FM achieves the highest
protection and fidelity trade offs, AttGAN provides the higher fidelity while
$WGAN{}$ has the highest protection.

###  16.4. Adult2 Census with results

Figure 32. Results on the perspective of WGAN (Adult2 dataset), which is the
transformation of data such that they resemble the privileged group.

On the WGAN perspective (Figure 32), we can observe that among all the trade-
offs achievable on the FM Pareto-front, there is at least one trade-off for
which the highest transformation $\textit{Pc}{}_{prot}$ is almost equal to
$1$. For GANSan and GANSan-OM, the highest classification
$\textit{Pc}{}_{prot}$ among all trade-offs is less than $0.5$, while it is
close to $0.97$ for $WGAN{}$. $AttGAN{}$ can achieve the perfect score of $1$.

Figure 33. Results on the perspective of AttGAN (Adult2 dataset), which is the
transformation of data such that they resemble the privileged group.

In the perspective of AttGAN in Figure 33, we can observe that $FM{}$ exhibits
an important negative slope with the increase of $Fid{}_{all}$. It is
impossible for all approaches to map the data onto the privileged distribution
while ensuring that the data are perfectly reconstructed. We can also note
that $FM{}$ dominates $AttGAN{}$ on fidelities lower than $0.989$. $AttGAN{}$
is better for higher fidelities. Both approaches seem to form a continuous
front, as we have observed in some other cases.

Figure 34. Results on the perspective of GANSan (Adult2 dataset), which is the
transformation of data such that they resemble the privileged group.

On the GANSan perspective (Figure 34), we can observe almost the same
performances as with the single sensitive attribute of GANSan.

FM can protect the sensitive information better than any other approach. The
fidelity values obtained are close to the perfect one of $Fid_{all}=1$ even
though the reconstruction constraint is only applied on the privileged group.
At the same time, GANSan, which reconstructs both groups achieves higher
fidelities (at most $1.27\%$ greater than the fidelities of $FM{}$), but lower
protection. This suggests that enforcing the reconstruction constraint only on
one group (the privileged group for instance) while ensuring the protection
with the original data version of the reconstructed group (i.e. the original
privileged group) provide a better structure for protecting sensitive
information. It is important to observe that $GANSan{}-OM$ failed to protect
the sensitive attribute. Thus, to better protect the sensitive attribute, some
degree of freedom (by not constraining the protected group) are necessary. FM
provides both of these requirements. That’s why the protection is higher. At
the same time, to avoid having the unconstrained group transformed into a
completely random distribution, we guide the transformation by ensuring the
mapping towards the privileged distribution.

##  17\. Standard deviation fairness results

This section presents the standard deviation results of the Table presented in
the main section of our article. Table 9 presents the standard deviation of
the fairness results, and Table 10 the standard deviation of the diversity.

Table 9. Cross validation protection results for all experimented datasets.
Standard deviation of fairness results presented in Table 5

| Adult | Adult2 | German | Lipton
---|---|---|---|---
|  | Baseline | FM | Baseline | FM | Baseline | FM | Baseline | FM
Metrics | $BER{}_{rc_{prv}}$ | $0.0022$ | $0.0298$ | $0.0073$ | $0.0409$ | $0.0215$ | $0.0480$ | $0.0318$ | $0.0202$
$BER{}_{og_{prv}}$ | $0.0022$ | $0.0312$ | $0.0073$ | $0.0232$ | $0.0215$ | $0.0325$ | $0.0318$ | $0.0166$
$SAcc{}_{rc_{prv}}$ | $0.0008$ | $0.0198$ | $0.0026$ | $0.0199$ | $0.0172$ | $0.0081$ | $0.0318$ | $0.0202$
$SAcc{}_{og_{prv}}$ | $0.0008$ | $0.0223$ | $0.0026$ | $0.0053$ | $0.0172$ | $0.0201$ | $0.0318$ | $0.0166$
$MI_{rc_{prv}}$ | $0.0048$ | $0.0275$ | $0.0070$ | $0.0361$ | $0.0120$ | $0.0182$ | $0.0461$ | $0.0068$
$MI_{og_{prv}}$ | $0.0048$ | $0.0753$ | $0.0070$ | $0.0171$ | $0.0120$ | $0.0296$ | $0.0461$ | $0.0120$
$\textit{Pc}{}_{prot}$ | $0.0000$ | $0.1241$ | $0.0000$ | $0.3640$ | $0.0000$ | $0.0972$ | $0.0000$ | $0.0877$
$Fid_{priv}$ | $0.0001$ | $0.0022$ | $0.0001$ | $0.0242$ | $0.0011$ | $0.0071$ | $0.0036$ | $0.0010$

Table 10. Standard deviation of the diversity

| Whole dataset | Protected group | Privileged group
---|---|---|---
Baseline | fairmapping | Baseline | Fairmapping | Baseline | Fairmapping
Adult2 | 0.0004 | 0.0221 | 0.0002 | 0.0301 | 0.0006 | 0.0192
Adult | 0.0002 | 0.0049 | 0.0003 | 0.0074 | 0.0005 | 0.0031
Lipton | 0.0046 | 0.0100 | 0.0103 | 0.0145 | 0.0014 | 0.0064
German | 0.0010 | 0.0062 | 0.0066 | 0.0096 | 0.0005 | 0.0069

##  18\. Fairness classification use case results

As presented in section 6.2, the fairness of our approach can be measure on
the fair classification use case.

In this setup, the objective is to build a classification mechanism that is
free from discrimination by using the transformed dataset to train a
classifier, while testing the model with original (unmodified) data. The
rationale of this scenario is to circumvent unusual or unethical issues that
are the result of the modifications introduced by the transformation procedure
(_e.g._ , to prevent the inference of groups, the value of the attribute
infraction, from a traffic ticket to a heinous crime), which would not be
accepted in critical domains such as predictive justice. Thus, by training the
classifier on transformed data in which the sensitive attribute is hidden, it
will learn to predict the decision without any information on the sensitive
attribute. By using the original test set on the trained classifier, the
practitioner avoids those risks of unethical outcomes.

Figure 35. Accuracies achieved with the MLP classifier on the Fair
classification scenario. The black vertical bar indicates the standard
deviation across all computed folds.  Figure 36. Accuracies computed in each
of the dataset group, for the fair classification scenario.

In Figure 35, we showcase the accuracy achieved with the fair classification
scenario. The fair classification accuracies are quite similar to those in the
other scenario. This suggests that training a classifier on either the
original or transformed and reconstructed dataset to predict the original
decision using either the original or the transformed and reconstructed
datasets will yield the same results. These datasets therefore share similar
characteristics which allow a classifier to behave similarly in all cases. The
group accuracy in Figure 36 also confirms our observation on the similarity of
predictions.

Figure 37. Fairness results for the fair classification.

Concerning the fairness results (Figure 37), they are not as significantly
improved as observed in the other scenario. As a matter of fact, we can
observe in Figure 38 that the metrics on the transformed sets follows the same
trend as the baseline.

Figure 38. Accuracy of the task decision at each group level for the scenario
data publishing.

##  19\. Metrics computed within each groups

In this section, we present the fairness and accuracies computed presented in
the main article at the group level basis.

Figure 39 present the accuracy of the task decision at each group level for
the scenario data publishing.

Figure 39. Accuracy of the task decision at each group level for the scenario
data publishing.

Figure 40 presents the fairness metrics computed in each group of the dataset

Figure 40. Accuracy of the task decision at each group level for the scenario
data publishing.

Figure 41 present the accuracy of the task decision at each group level for
the scenario Local sanitization.

Figure 41. Accuracy of the task decision at each group level for the scenario
local sanitization.

Figure 41 presents the fairness metrics computed in each group of the dataset

Figure 42. Accuracy of the task decision at each group level for the scenario
data publishing.

##  20\. Lower dimensional visualisation of distributions

In this appendix, we present some lower dimensional visualisation of our
results.

Figure 43. Lower dimensional plot of the original Lipton dataset. Figure 44.
Lower dimensional plot of the Lipton dataset with the transformed protected
group data. Figure 45. Lower dimensional plot of the Lipton dataset with the
transformed protected data and the reconstructed data from the privileged
group.

In Figure 43 we present the lower dimensional plot of the original Lipton
dataset, while 45 represent the lower dimensional of the transformed protected
group and the reconstructed privileged data. Figure 44 present the same plot,
but with the original privileged distribution. We can observe that the
transformed protected distribution overlap the privileged distribution, which
means that the transformation is successful.

Figure 46. Lower dimensional plot of the original Lipton dataset. Figure 47.
Lower dimensional plot of the Lipton dataset with the transformed protected
group data.

On German Credit (Figure 46 and Figure 47) the transformation impact is less
visible since the original protected distribution is not easily
distinguishable from the privileged one. Nevertheless, we can observe a slight
improvement in the location of the protected distribution, which is more
embedded within the target distribution.

Figure 48. Lower dimensional plot of the original Lipton dataset. Figure 49.
Lower dimensional plot of the Lipton dataset with the transformed protected
group data.

The results on the dataset Adult shows an improvement on the superposition of
the transformed protected distribution onto the privileged one (figures 48,
49).

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