=Paper= {{Paper |id=Vol-3276/SSS-22_FinalPaper_69 |storemode=property |title=Fairness-aware Naive Bayes Classifier for Data with Multiple Sensitive Features |pdfUrl=https://ceur-ws.org/Vol-3276/SSS-22_FinalPaper_69.pdf |volume=Vol-3276 |authors=Stelios Boulitsakis-Logothetis |dblpUrl=https://dblp.org/rec/conf/aaaiss/Boulitsakis-Logothetis22 }} ==Fairness-aware Naive Bayes Classifier for Data with Multiple Sensitive Features== https://ceur-ws.org/Vol-3276/SSS-22_FinalPaper_69.pdf

Fairness-aware Naive Bayes Classifier for Data with Multiple Sensitive Features

Stelios Boulitsakis-Logothetis
University of Durham
Durham, United Kingdom
stelios.b.logothetis@gmail.com

Abstract tion here in Table 1. Traditionally, the proposed notions have
been classified into two categories. The simplest and most
Fairness-aware machine learning seeks to maximise utility in well-studied, group fairness, is based on defining distinct
generating predictions while avoiding unfair discrimination
based on sensitive attributes such as race, sex, religion, etc.
protected groups in the given data. Then, for each of these
An important line of work in this field is enforcing fairness groups, a user-selected statistical constraint must be satis-
during the training step of a classifier. A simple yet effec- fied. This has notable disadvantages: It requires groups to be
tive binary classification algorithm that follows this strategy treated fairly in aggregate, but this guarantee does not nec-
is two-naive-Bayes (2NB), which enforces statistical parity - essarily extend to individuals (Awasthi et al. 2020). Further,
requiring that the groups comprising the dataset receive pos- different statistical constraints prioritise different aspects of
itive labels with the same likelihood. In this paper, we gen- fairness. Many of them have also been shown to be incom-
eralise this algorithm into N-naive-Bayes (NNB) to eliminate patible with each other, making the choice even more diffi-
the simplification of assuming only two sensitive groups in cult for users. Finally, the choice of the protected groups that
the data and instead apply it to an arbitrary number of groups. should be considered is an open question (Blum et al. 2018;
We propose an extension of the original algorithm’s statistical Kleinberg, Mullainathan, and Raghavan 2017).
parity constraint and the post-processing routine that enforces
statistical independence of the label and the single sensitive
An orthogonal notion to group fairness is individual fair-
attribute. Then, we investigate its application on data with ness. Put simply, this notion requires that ”similar individu-
multiple sensitive features and propose a new constraint and als be treated similarly” (Dwork et al. 2012). This approach
post-processing routine to enforce differential fairness, an ex- addresses the previous lack of any individual-level guaran-
tension of established group-fairness constraints focused on tees. However, it requires strong functional assumptions and
intersectionalities. We empirically demonstrate the effective- still requires the step of choosing an underlying metric over
ness of the NNB algorithm on US Census datasets and com- the dataset features (Awasthi et al. 2020).
pare its accuracy and debiasing performance, as measured by Alternative models of fairness have been proposed to ad-
disparate impact and DF-ϵ score, with similar group-fairness
dress the disadvantages of the two traditional definitions.
algorithms. Finally, we lay out important considerations users
should be aware of before incorporating this algorithm into One model is causal fairness, which examines the unfair
their application, and direct them to further reading on the causal effect the sensitive attribute value may have on the
pros, cons, and ethical implications of using statistical parity prediction made by an algorithm (Mhasawade and Chunara
as a fairness criterion. 2021). Another, which is explored in this paper, is differ-
ential fairness (DF). This is an extension of the established
group fairness concepts that applies them to the case of inter-
1 Introduction sectionalities, meaning groups that are defined by multiple
Today, countless machine learning-based systems are in use overlapping sensitive attributes (Foulds et al. 2020; Morina
that autonomously make decisions or aid human decision- et al. 2019).
makers in applications that significantly impact individu- A similar model is statistical parity subgroup fairness
als’ lives. This has made it vital to develop ways of en- (SF), which focuses on mitigating intersectional bias by ap-
suring these models are trustworthy, ethical, and fair. The plying group fairness to the case of infinitely many, very
field of fairness-aware machine learning is centered on en- small subgroups (Kearns et al. 2018). SF and DF are no-
hancing the fairness, explainability, and auditability of ML table because they both enable a more nuanced understand-
models. A goal many research works in this field share is ing of unfairness than when a single sensitive attribute and
to maximise utility in generating predictions while avoiding broad, coarse groups are considered. A key difference be-
discrimination against people based on specific sensitive at- tween them, however, is DF’s focus on minority groups. The
tributes, such as race, sex, religion, nationality, etc. SF measure of subgroup parity weighs larger groups more
Researchers have devised many formalisations to try and heavily than very small ones, while DF-parity considers all
capture intuitive notions of fairness, each with different pri- groups equally. This means DF can provide greater protec-
orities and limitations. We summarise the ones we will men- tion to very small minority groups since, in SF, their impact
___________________________________
In T. Kido, K. Takadama (Eds.), Proceedings of the AAAI 2022 Spring Symposium
“How Fair is Fair? Achieving Wellbeing AI”, Stanford University, Palo Alto, California,
USA, March 21–23, 2022. Copyright © 2022 for this paper by its authors. Use permitted
under Creative Commons License Attribution 4.0 International (CC BY 4.0).

56
on the overall score is reduced (Foulds et al. 2020). Name Definition
Despite the lack of consensus on any universal notion of
fairness, research has proceeded using the existing models. Statistical Likelihood of positive prediction given group
A major line of work in the development of fair learning al- Parity membership should be equal for all groups.
gorithms is enforcing fairness during the training step of a Disparate Mean ratio of positive predictions for each pair
classifier (Donini et al. 2018). A simple yet effective algo- Impact of groups should be 1 or greater than p%.
rithm that follows this strategy is Calders and Verwer’s two-
naive-Bayes algorithm (Calders and Verwer 2010) (2NB). Subgroup Group fairness applied to infinite number of
This algorithm was originally proposed as one of three ways Fairness very small groups.
of pursuing fairness in naive Bayes classification. It received Differential Group fairness applied to groups defined by
further attention in the 2013 publication (Kamishima et al. Fairness multiple overlapping sensitive attributes.
2013) which asserted its effectiveness in enforcing group
fairness in binary classification and explored its underlying Individual Distance between the likelihood of outcomes
statistics. It works by training separate naive Bayes classi- Fairness between any two individuals should be no
fiers for each of the two (by assumption) groups comprise greater than similarity distance between them.
the dataset, the privileged and the non-privileged group. Causal Use of causal modelling to find effect of sensi-
Then, the algorithm iteratively assesses the fairness of the Fairness tive attributes on predictions.
combined model and makes small changes to the observed
probabilities in the direction of making them more fair Table 1: Some notable formalisations of fairness.
(Friedler et al. 2019).
A recent publication exploring the arguments for and
against statistical parity (Räz 2021) has served as motiva- Related Work
tion to re-visit algorithms based around it. Statistical parity
(also referred to as demographic parity or independence) is Naive Bayes Naive Bayes is a probabilistic data mining
a group fairness notion which requires that the groups com- and classification algorithm. In spite of its relative simplic-
prising the dataset receive positive labels with the same like- ity, it has been shown to be very competent in real-world ap-
lihood. An assumption that is at the core of 2NB and many plications that require classification or class probability esti-
other research works, however, is that of a single, binary mation and ranking1 . Various strategies have been explored
sensitive feature (Oneto, Donini, and Pontil 2020). This as- for improving the algorithm’s performance by weakening its
sumption has been noted to rarely hold in the real world, and conditional independence assumption. These include struc-
eliminating it is one of the essential goals of the previously ture extension, attribute weighting, etc. These techniques
introduced notions of differential fairness and subgroup par- focus on maximising accuracy or averaged conditional log
ity fairness (Foulds et al. 2020; Kearns et al. 2018). likelihood (Jiang 2011). Calders and Verwer’s proposal of
This opens the question of how 2NB can be applied to composing multiple naive Bayes models instead aims to en-
data with multiple, overlapping sensitive attributes while force independence of predictions with respect to a binary
avoiding oversimplification. The 2NB algorithm is applica- sensitive feature, thus satisfying the statistical parity con-
ble to a wide range of tasks and its effectiveness, even in straint between the two groups (Calders and Verwer 2010).
comparison to more complex algorithms, has been demon-
Fair Classification There is a large body of research into
strated (Kamishima et al. 2013; Friedler et al. 2019). At the
designing learning methods that do not use sensitive infor-
same time, its’ design is sufficiently elegant and intuitive to
mation in discriminatory ways (Oneto, Donini, and Pontil
be approachable to practitioners across many disciplines - an
2020). As mentioned, various formalisations of fairness ex-
important advantage. Thus, extending the algorithm to cover
ist but the most well-studied one is group fairness (Blum
more use cases will be the focus of this work.
et al. 2018). Many algorithms designed around this notion
Contributions are introduced as part of the comparative experiment in Sec-
tion 3.
This paper seeks to build upon Calders and Verwer’s work A more recent proposal, differential fairness (DF), ex-
by exploring the following: tends existing group fairness concepts to protect subgroups
• We adapt the original 2NB structure and balancing rou- defined by intersections of and by individual sensitive at-
tine to support multiple, polyvalent (categorical) sensi- tributes. The original papers by (Foulds et al. 2020) and
tive features. (Morina et al. 2019) explore the context of intersectionality,
• We use this new property of the algorithm to apply it to and provide comparisons of DF with established concepts.
differential fairness. The first paper asserts DF’s distinction from subgroup parity
• To support the above, we examine the extended algo- and demonstrates its usefulness in protecting small minority
rithm’s performance on real-world US Census data. groups. The latter paper gives methods to robustly estimate
the DF metrics and proposes a post-processing technique to
• Finally, we lay out important considerations users should enforce DF on classifiers.
be aware of before using this algorithm. We draw upon
the literature to lay out the pros, cons, and ethical impli- 1
Recent, novel applications include (Valdiviezo-Diaz et al.
cations of using statistical parity as a fairness criterion. 2019; Feng et al. 2018; Niazi et al. 2019) among others.

57
Humanistic Analysis A line of work that is parallel to final predicted class probabilities, for a sample xs (where x
fair algorithm development focuses on analysing these pro- is the feature vector excluding the sensitive feature s), is:
posals from an ethical, philosophical, and moral standpoint.
A recent such publication, which examines statistical par-
ity among other notions, and which motivated and influ- P (y|xs) = P (x|y) ∗ P (s|y) ∗ P (y) (2)
enced this paper, is by Hertweck, Heitz, and Loi (Her- = Cs (x) ∗ P (s|y) ∗ P (y) (3)
tweck, Heitz, and Loi 2021). They propose philosophically- = Cs (x) ∗ P (s ∩ y) (4)
grounded criteria for justifying the enforcement of indepen-
dence/statistical parity in a given task. They include scenar- Where Cs is the the sub-estimator for sensitive group s ∈ S.
ios where enforcing statistical parity is ethical and justified,
as well as counter-examples where the criteria are met but
Enforcing Statistical Parity
independence should not be enforced. As with many sim- To satisfy the statistical parity constraint, the original 2NB
ilar works, they conclude by directing the reader to strike algorithm runs a heuristic post-processing routine that iter-
a balance between fairness and utilitarian concerns (such as atively adjusts the conditional probabilities P (Y |S) of the
accuracy) in their task. (Heidari et al. 2019) do similar work, groups in the direction of making them equal. During its
laying out the moral assumptions underlying several popu- execution, this probability-balancing routine alternates be-
lar notions of fairness. In (Räz 2021), Räz critically exam- tween reducing N (Y = 1, S = 1) and increasing N (Y =
ines the advantages and shortcomings of statistical parity as 1, S = 0) depending on the number of positive labels out-
a fairness criterion and makes an overall positive case for it. putted by the model at each iteration. This is to try and keep
(Friedler, Scheidegger, and Venkatasubramanian 2016) the resultant marginal distribution of Y stable. Once balanc-
introduce the concept of distinct worldviews which influence ing is complete, the value of P (S|Y ) can be induced from
how we pursue fairness. One of them is that We’re All Equal Ny,s similar to (1). The first contribution of this paper is to
(WAE) i.e. there is no association between the construct (the extend this routine to suit the polyvalent definition of statis-
latent feature that is truly relevant for the prediction) and the tical parity we will use:
sensitive attribute. The orthogonal worldview is that What Definition 1. Statistical (Conditional) Parity for Polyvalent
You See Is What You Get, wherein the observed labels are S (Ritov, Sun, and Zhao 2017):
accurate reflections of the construct. In (Yeom and Tschantz For predicted binary labels ŷ and polyvalent sensitive fea-
2021), Yeom and Tschantz give a measure of disparity am- ture S, statistical (conditional) parity requires 3 :
plification and dissect the popular group fairness models of
statistical parity, equalised odds, calibration, and predictive P (ŷ = 1|s) = P (ŷ = 1|s′ ) ∀ s, s′ ∈ S (5)
parity through the lens of worldviews. They argue that un- We modify the probability-balancing routine to subtract
der WAE, statistical parity is required to eliminate disparity and add probability to the group with the highest (max) and
amplification. However, deviating from this worldviews in- lowest (min) current P (Y = 1|s) respectively. These prob-
troduces inaccuracy when we enforce parity. abilities are re-computed with each iteration, and the max
and min groups re-selected. Further, we introduce the con-
2 N-Naive-Bayes Algorithm straint that only groups designated by the user as privileged
can receive a reduction in their likelihood of getting a posi-
The proposed N-naive-Bayes algorithm is a supervised bi- tive label 4 . This is to avoid making any assumptions about
nary classifier that allows the enforcement of a statistical which groups it would be appropriate to demote positive
fairness constraint in its predictions. Given an (ideally large) instances of. It allows the balancing routine to terminate
training set of labelled instances, the algorithm partitions the immediately if it over-corrects, or if the data is such that
data based on sensitive attribute value and trains a separate P (ŷ = 1|snp ) > P (ŷ = 1|sp ) to begin with, as is the case in
naive Bayes sub-estimator on each of the sub-sets. This is an the well-known UCI Adult dataset, for example. This gives
extension of the original two-naive-Bayes structure, where us the final form of our statistical parity criterion:
exactly two sub-estimators are trained. The next step of the
training stage is for the conditional probabilities P (Y |S) to Definition 2. Statistical Parity Criterion for NNB:
be empirically estimated from the training set. Where Ns is For predicted binary labels ŷ and sensitive feature S:
the number of instances that belong to group s, and Ny,s P (ŷ = 1|sp ) = P (ŷ = 1|snp ) ∀ (sp , snp ) ∈ Sp × Snp (6)
the number of instances of that group that have label y, the
empirical conditional probability2 is given as: Where Sp and Snp are the sub-sets of all privileged and non-
privileged sub-groups of S respectively.
Ny,s + α We adapt the above definition into a score that the algo-
P (y|s) = (1)
Ns + 2 ∗ α rithm can minimise:
Finally, the algorithm modifies the joint distribution
P (Y, S) to enforce the given fairness constraint. Then, the disc = max P (ŷ = 1|sp ) − min P (ŷ = 1|snp ) (7)
2 3
Equation (1) gives a smoothed empirical probability, where the The cited definition requires this to hold for all values of ŷ,
constant α is the parameter of a symmetric Dirichlet prior with however for a binary label it is sufficient to check ŷ = 1.
4
concentration parameter 2 ∗ α, since a binary label is assumed. A similar constraint is explored by (Zafar et al. 2017).

58
Algorithm 1: Pseudocode for a probability-balancing routine This is used in experiments to estimate the value of ϵ (the
to enforce statistical parity ϵ-score) from the predicted labels on the dataset5 . In exper-
1: Calculate the parity score, disc, of the predicted classes iments we set β = 2 ∗ α and substitute with the observed
by the current model and store smax , smin conditional probability estimates from the dataset. An addi-
2: while disc > disc0 do tional measure given in (Foulds et al. 2020) to assess fair-
3: Let numpos be the number of positive samples by ness from the standpoint of intersectionality is differential
the current model fairness bias amplification. This measure gives an indication
4: if numpos < the number of positive samples in the of how much a black-box classifier increases the unfairness
training set then over the original data (Foulds et al. 2020; Zhao et al. 2017).
5: N (y = 1, smin ) + = ∆ ∗ N (y = 0, smin ) Definition 5. Differential Fairness Bias Amplification
6: N (y = 0, smin ) − = ∆ ∗ N (y = 0, smin ) A classifier C satisfies (ϵ2 − ϵ1 )-DF bias amplification
7: else w.r.t. dataset D if C is ϵ2 -DF fair and D is ϵ1 -DF fair.
8: N (y = 1, smax ) + = ∆ ∗ N (y = 1, smax ) To adjust the joint distribution P (Y, S) to minimise sat-
9: N (y = 0, smax ) − = ∆ ∗ N (y = 1, smax ) isfy DF-fairness and minimise the ϵ-score, we propose a
10: end if new heuristic probability-balancing routine and associated
11: If any N (y, s) is now negative, rollback the changes discrimination score. The distinction from the balancing
and terminate routine given in Algorithm 1 is that this focuses on out-
12: Recalculate P (Y |S), disc, smax , smin putting a narrower range of probabilities, while still avoid-
13: end while ing negatively impacting groups that are designated as non-
privileged. To form the new discrimination score, we ap-
ply the principle of separating privileged and non-privileged
Note that the above criterion can easily be relaxed to ap-
sub-groups of S from the previous section to the ϵ-score def-
ply the four-fifths rule for removing disparate impact (or its
inition:
more general form, the p% rule (Zafar et al. 2017)) instead
of perfect statistical parity. For the purposes of this paper,
however, we explore the effect of statistical parity in its base P (ŷ = 1|snp )
e−ϵ ≤ ≤ eϵ ∀ (sp , snp ) ∈ Sp × Snp (10)
form. P (ŷ = 1|sp )
We also note the definition of disparate impact we use in
the evaluation stage: We then express this restricted ϵ-score as the maximum of
two ratios: eϵ = max(ρd , ρu ), where for (sp , snp ) ∈ Sp ×
Definition 3. Disparate Impact (Mean) for Polyvalent S: Snp :
1 X P (ŷ = 1|snp )
|Sp × Snp | P (ŷ = 1|sp ) P (ŷ = 1|snp ) P (ŷ = 1|sp )
(sp ,snp )
ρd = max , ρu = max (11)
Algorithm 1 describes the extended probability balanc- P (ŷ = 1|sp ) P (ŷ = 1|snp )
ing heuristic for enforcing parity. The values of sp , snp in The execution of the proposed balancing routine is de-
the parity criterion (Equation 7) are referred to as smax and termined by these ratios. If ρd is greater, then the non-
smin respectively. At each iteration, the routine determines privileged sub-group with smallest probability at that itera-
these groups and adjusts their conditional probabilities. A tion receives an increase in probability. If ρu is greater, then
further modification from the original is that the proportion the privileged group with highest probability receives a de-
by which the probabilities are adjusted with each iteration is crease in probability. These conditions can be expected to
now proportional to the size of the group itself, instead of the alternate as the conditional probabilities P (Y |S) converge.
size of the opposite group. In experiments, this yields a great Iteration continues until ρd is close to zero. The smax and
performance improvement, especially where the distribution smin groups are determined as in the previous section.
of samples over S is very imbalanced. This routine disregards the number of positive labels the
model produces, while Algorithm 1 attempts to keep that
Enforcing Differential Fairness number close to the number of positive labels in the train-
An alternative measure of fairness we explore is differential ing data. This allows it to avoid situations where a single,
fairness, as given in (Foulds et al. 2020). non-privileged sub-group with small probability would re-
Definition 4. A classifier is ϵ-differentially fair if: quire the probabilities of the privileged groups to be reduced
significantly. In such cases, other non-privileged sub-groups
P (ŷ|s) might maintain much higher probabilities, therefore giving
e−ϵ ≤ ≤ eϵ ∀ s, s′ ∈ S, ŷ ∈ Y (8)
P (ŷ|s′ ) a poor ϵ-score. An further difference is the proportion by
The (smoothed) empirical differential fairness score, from 5
the empirical counts in the data, assuming a binary label, is: Note that this definition produces noisier estimates for sub-
groups with fewer members. (Morina et al. 2019) shows that as the
dataset grows, the given estimate converges to the true value, and
N (ŷ, s) + α N (s′ ) + β
e−ϵ ≤ ≤ eϵ ∀ s, s′ ∈ S, ŷ ∈ Y that this happens regardless of the chosen smoothing parameters.
N (s) + β N (ŷ, s′ ) + α However, for small or imbalanced datasets, more robust estimation
(9) methods should be used.

59
Algorithm 2: Pseudocode for a probability-balancing routine indicated after its name, e.g. Income-Race-Sex is the
to enforce DF parity Income task using race and sex as the sensitive features.
1: Calculate the ratios ρd , ρu empirically from the pre- To best capture intersectional fairness when using multiple
dicted classes by the current model, store smax , smin sensitive features, we follow the approach from (Foulds et al.
2: while ρd > disc0 do 2020) and define each group s as a tuple of the sub-groups
3: if ρu ≤ ρd then of each sensitive feature that each sample belongs to.
4: N (y = 0, smin ) − = ∆ ∗ N (y = 0, smin ) First Experiment This experiment compares NNB’s per-
5: N (y = 1, smin ) + = ∆ ∗ N (y = 1, smin ) formance with other algorithms. The comparison includes
6: else ”vanilla” models as baselines for performance, and several
7: N (y = 0, smax ) + = ∆ ∗ N (y = 0, smax ) group-fairness-aware algorithms that have a similar focus to
8: N (y = 1, smax ) − = ∆ ∗ N (y = 1, smax ) NNB - ensuring non-discrimination across protected groups
9: end if by optimising metrics such as statistical parity or disparate
10: Recalculate P (Y |S), ρd , ρu , smax , smin impact. Specifically, we consider the following:
11: end while
• GaussianNB, DecisionTree, LR, SVM: scikit-
Learn’s Gaussian naive Bayes, Decision Trees, Logistic
Regression, and SVM.
which each Ny,s is modified grows/decreases exponentially.
In experiments, this allows the routine escape local minima • Feldman-DT, Feldman-NB: A pre-processing algo-
that occur during the adjustment of P (Y |S) and lead to in- rithm that aims to remove disparate impact. It equalises
efficiency. This routine does, however, offer a theoretical ac- the marginal distributions of the subsets of each attribute
curacy trade-off compared to Algorithm 1, which we inves- with each sensitive value (Feldman et al. 2015). The re-
tigate in the following section. sulting ”repaired” data is then used to train scikit-Learn
Finally, note that all the above probability-balancing rou- classifiers - Decision Trees (DT) and Gaussian naive
tines (including Calders and Verwer’s original one) are Bayes (NB).
based around the assumption that the distribution of labels • Kamishima: An in-processing method that introduces a
over the sensitive feature(s) in the training set is reflective of regularisation term to logistic regression to enforce inde-
the test setting. This assumption is not unique to this model pendence of labels from the sensitive feature (Kamishima
(see (Agarwal et al. 2018; Hardt, Price, and Srebro 2016)), et al. 2012).
and under it, we can conclude that minimising the given fair- • ZafarAccuracy, ZafarFairness: An in-
ness measure on the training set generalises to the test data processing algorithm that applies fairness constraints
(Singh et al. 2021). to convex margin-based classifiers (Zafar et al. 2017) .
Specifically, we test two variations of a modified logistic
3 Experimental Results regression classifier: The first maximises accuracy
Setup subject to fairness (disparate impact) constraints, while
the latter prioritises removing disparate impact.
We implement NNB in Python within the scikit-Learn
• 2NB: Calders and Verwer’s original algorithm, using the
framework, using Gaussian naive Bayes as the sub-
same GaussianNB sub-estimator as NNB.
estimator. We then evaluate its performance in two experi-
ments. • NNB-Parity, NNB-DF: N-naive-Bayes tuned to sat-
For both experiments, we use real-world data from the US isfy statistical parity using Algorithm 1, and DF-parity
Census Bureau6 . (Ding et al. 2021) define several classifica- using Algorithm 2.
tion tasks on this data, each involving a sub-set of the total For the comparison we use the benchmark provided by
features available. We consider two: (Friedler et al. 2019). The fairness-aware algorithms are
• Income: Predict whether an individual’s income is tuned via grid-search to optimise accuracy. The performance
above $50,000. The data for this problem is filtered so of the algorithms is then measured over ten random train-test
that it serves as a comparable replacement to the well- splits of the data.
known UCI Adult dataset. Second Experiment This experiment demonstrates how
• Employment: Predict whether an individual is em- NNB performs in finer detail. We consider GaussianNB,
ployed NNB-Parity, and NNB-DF as before, and we further in-
The details of which features are included in each task and clude 2NB, the original two-naive-Bayes algorithm imple-
what filtering takes place can be found in the paper (Ding mented identically to NNB. Finally, we include Perfect
et al. 2021) and the associated page on GitHub7 . To eval- as a secondary baseline, to illustrate the scores that would
uate NNB we use data from the 2018 census in the state be achieved by a perfect classifier.
of California. The sensitive feature(s) used in each task are To evaluate the performance of the above algorithms, we
note the mean and variance of the following measures over
6
https://www.census.gov/programs- 10 random train-test splits: accuracy, AUC, disparate im-
surveys/acs/microdata/documentation.html pact score (mean of the DI between all privileged and non-
7
https://github.com/zykls/folktables privileged groups), statistical parity score (as defined in 2),

60
Figure 1: Scatter plots of accuracy vs. disparate impact for Income-Race and vs. ϵ-score for Income-Race-Sex

DF-ϵ (as defined in 4), DF-bias amplification score (as de- On Employment-Race all naive Bayes models achieve
fined in 5). We also compare the resultant distribution of la- similar accuracy, while DT and LR-based models rank
bels over groups of S on a single random train-test split. higher, and SVM the highest. The same can be observed for
Employment-Race-Sex, and for both tasks NNB-DF again
Results gives the ϵ-scores closest to zero.

First Experiment Figure 1 gives the accuracy vs. the Second Experiment Table 2 gives the scores achieved
disparate impact and DF-ϵ scores on the Income-Race on the Income-Race task, and Table 3 gives the
and Income-Race-Sex tasks. Figure 2 shows the same for same Employment-Race-Sex. On Income-Race,
Employment-Race and Employment-Race-Sex. It can be both NNB models gave an improved parity score compared
seen that on Income-Race, NNB results in a higher DI score to the perfect classifier and GaussianNB. NNB and 2NB
than 2NB and has often over-favoured non-privileged groups also gave improved disparate impact scores over the baseline
causing a score > 1. Its accuracy is on-par with 2NB and models, but 2NB under-corrected while the NNB models
the baseline naive Bayes, DT, and LR models. Feldman’s al- gave a score > 1 indicating they favoured the non-privileged
gorithm with Decision Trees results similar disparate impact groups over the privileged group.
score in some splits, but lower accuracy. The same is true for NNB-Parity and NNB-DF gave similar disparate im-
the DF-ϵ score on this task. On Income-Race-Sex, NNB-DF pact scores, but the former gave higher accuracy while the
beats out all other algorithms in achieving DI ∼ 1, however latter produced a narrower range of positive label propor-
NNB-Parity has higher accuracy than both NNB-DF and tions, and thus better parity, ϵ, and DF-bias amplification
naive Bayes. NNB-DF is also the most successful at min- scores. The evident accuracy trade-off is more pronounced
imising the ϵ-score for this task, though again this comes at in the latter task, with NNB-Parity achieving an accuracy
the cost of lower accuracy than the baseline model. of 0.7445 ± 0.00, and NNB-DF achieving 0.7199 ± 0.00.

61
Figure 2: Scatter plots of accuracy vs. disparate impact for Employment-Race and vs. ϵ-score for Employment-Race-Sex

On Employment-Race-Sex, NNB-DF outperformed Statistical Parity as a Fairness Criterion Statistical par-
NNB-Parity on all scores. This was also the case for ity stands opposed to the (aggregate) accuracy of a classifier,
Employment-Race, where both models had similar ac- except in degenerate cases where the data is already fair, so
curacy but NNB-DF displayed less over-correction in its it is recommended that a balance between the two is pursued
disparate impact score (1.0336 ± 0.0001 versus 1.2760 ± (Hertweck, Heitz, and Loi 2021). This also applies to the ex-
0.0002), in addition to the expected improvement in ϵ-score tended, but still parity-based, DF measure that was explored
(0.1068 ± 0.001 versus 0.3434 ± 0.0001). This suggests the in Section 2. In their worldview-based analysis, Yeom and
DF balancing routine is better suited for the Employment Tschantz caution us that even under WAE, blind enforce-
task than the parity-based routine. ment of statistical parity can introduce new discrimination
into the system (Yeom and Tschantz 2021). Thus, users must
4 Discussion be aware of the ethical implications of using parity as a core
fairness constraint, the possible impact it may have on in-
In this work we presented an extension of the two-naive- dividuals, and the moral objections these individuals may
Bayes algorithm, adapting it to suit datasets with multi- justifiably raise.
ple, polyvalent sensitive features. We applied the proposed
N-naive-Bayes structure to intersectionality and differen-
tial fairness by giving an alternative probability-balancing We recommend further reading on the advantages and dis-
routine. Our experiments on real-world datasets yielded advantages of group fairness in general (Räz 2021; Dwork
favourable results and demonstrated the effectiveness and et al. 2012; Heidari et al. 2019), as well as parity specifically
the differences between the parity and DF-based approaches. (Hertweck, Heitz, and Loi 2021; Yeom and Tschantz 2021),
We conclude by laying out key considerations users so users can make informed decisions on how to apply sta-
should take into account before using N-naive-Bayes: tistical parity and N-naive-Bayes to their application.

62
AUC Accuracy DI Parity DF-ϵ DF-amp
GaussianNB 0.8270 ± 0.00 0.7503 ± 0.00 0.6304 ± 0.0001 0.4222 ± 0.0000 1.4100 ± 0.0012 0.4680 ± 0.0045
2NB 0.8223 ± 0.00 0.7577 ± 0.00 0.8930 ± 0.0013 0.3606 ± 0.0000 0.9774 ± 0.0016 0.0353 ± 0.0059
NNB-Parity 0.8114 ± 0.00 0.7480 ± 0.00 1.0810 ± 0.0006 0.1984 ± 0.0005 0.4580 ± 0.0045 −0.4840 ± 0.0041
NNB-DF 0.8138 ± 0.00 0.7380 ± 0.00 1.0636 ± 0.0007 0.1530 ± 0.0008 0.3112 ± 0.0048 −0.6308 ± 0.0035
Perfect 1.0000 ± 0.00 1.0000 ± 0.00 0.6975 ± 0.0005 0.2950 ± 0.0001 0.9420 ± 0.0048 0.0000 ± 0.0000

Table 2: Scores Achieved on Income with Race as the Sensitive Feature

AUC Accuracy DI Parity DF-ϵ DF-amp
GaussianNB 0.8159 ± 0.00 0.7273 ± 0.00 1.0228 ± 0.0001 0.3000 ± 0.0001 0.4994 ± 0.0003 0.0922 ± 0.0016
2NB 0.8112 ± 0.00 0.7202 ± 0.00 0.9352 ± 0.0001 0.2951 ± 0.0001 0.4818 ± 0.0002 0.0746 ± 0.0015
NNB-Parity 0.7820 ± 0.00 0.7241 ± 0.00 1.2990 ± 0.0004 0.2478 ± 0.0007 0.3971 ± 0.0013 −0.0101 ± 0.0005
NNB-DF 0.7909 ± 0.00 0.7251 ± 0.00 1.0601 ± 0.0002 0.1272 ± 0.0009 0.1840 ± 0.0018 −0.2232 ± 0.0011
Perfect 1.0000 ± 0.00 1.0000 ± 0.00 0.8643 ± 0.0001 0.1782 ± 0.0002 0.4072 ± 0.0014 0.0000 ± 0.0000

Table 3: Scores Achieved on Employment with Race and Sex as the Sensitive Features

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