# Bt-GAN: Generating Fair Synthetic Healthdata via Bias-transforming
Generative Adversarial Networks

Resmi Ramachandranpillai<EMAIL_ADDRESS>
Md Fahim Sikder<EMAIL_ADDRESS>
David Bergström<EMAIL_ADDRESS>
Fredrik Heintz<EMAIL_ADDRESS>
Department of Computer and Information Science (IDA),
Linköping University, Sweden

###### Abstract

Synthetic data generation offers a promising solution to enhance the
usefulness of Electronic Healthcare Records (EHR) by generating realistic de-
identified data. However, the existing literature primarily focuses on the
quality of synthetic health data, neglecting the crucial aspect of fairness in
downstream predictions. Consequently, models trained on synthetic EHR have
faced criticism for producing biased outcomes in target tasks. These biases
can arise from either spurious correlations between features or the failure of
models to accurately represent sub-groups. To address these concerns, we
present Bias-transforming Generative Adversarial Networks (Bt-GAN), a GAN-
based synthetic data generator specifically designed for the healthcare
domain. In order to tackle spurious correlations (i), we propose an
information-constrained Data Generation Process (DGP) that enables the
generator to learn a fair deterministic transformation based on a well-defined
notion of algorithmic fairness. To overcome the challenge of capturing exact
sub-group representations (ii), we incentivize the generator to preserve sub-
group densities through score-based weighted sampling. This approach compels
the generator to learn from underrepresented regions of the data manifold. To
evaluate the effectiveness of our proposed method, we conduct extensive
experiments using the Medical Information Mart for Intensive Care (MIMIC-III)
database. Our results demonstrate that Bt-GAN achieves state-of-the-art
accuracy while significantly improving fairness and minimizing bias
amplification. Furthermore, we perform an in-depth explainability analysis to
provide additional evidence supporting the validity of our study. In
conclusion, our research introduces a novel and professional approach to
addressing the limitations of synthetic data generation in the healthcare
domain. By incorporating fairness considerations and leveraging advanced
techniques such as GANs, we pave the way for more reliable and unbiased
predictions in healthcare applications.

##  1 Introduction

Clinical Decision Support Systems (?) are important for healthcare
organizations to improve care delivery in the era of value-based healthcare,
digital innovation, and big data. The adoption of advanced artificial
intelligence technology in healthcare systems is gathering interest from many
researchers (?), (?). One example is called precision medicine – predicting
which treatment procedures are likely to succeed on a patient based on
numerous traits and the treatment context – is the most prevalent application
of classical machine learning in healthcare. These systems can yield benefits
in terms of improved accuracy, diagnosis, finding new knowledge about the
illness conditions and their progression, and the ability to provide patients
with a more concrete prognosis and treatment plan by the use of historical
data in model development.

Data analysis on Electronic Healthcare Records (EHR) (?) is greatly affected
by rules such as the Health Insurance Portability and Accountability Act
(HIPPA) (?) in the US and the General Data Protection Regulation (GDPR) (?) in
EU, that preserve patient’s privacy. Healthcare organizations are now jointly
accountable for any personal data breach since they are responsible for
managing all personal data storage and processing in both their organization
and that of their suppliers. Over the last three years, Privacy Impact
Assessments (PIAs) (?) have become commonplace in the healthcare industry.
GDPR has now placed the delivery of PIAs in the public domain, boosting
transparency and information risk ownership clarity.

Synthetic data generation (?, ?) tries to preserve the privacy of patients by
producing realistic samples to perform downstream tasks. Generative
Adversarial Networks have drawn much attention, however, are often criticized
for producing low-quality and less diverse samples(?). The state-of-the-art
(SOTA) synthetic healthcare data generation methods (?, ?) have explored the
concepts of accuracy, utility, and privacy, but paid less attention to
fairness in the Data Generation Process (DGP) and in the subsequent tasks. To
facilitate the development of equitable analysis and predictions, the DGP
should ensure that the synthetic EHR is fair along with other dimensions (such
as utility, privacy, etc.).

There are malignant feature correlations in the medical data, which we call
correlation biases. GANs amplify these spurious correlations as studied in
(?). Another cause of unfairness can be in the direction of fair resemblance -
accurately capturing the diversities in sub-groups. We would like to mention
(?), as it raises some concerns about the trustworthiness of synthetic data in
the direction of fair resemblance. We use the term representation fairness
instead of fair resemblance throughout this study as this refers to how
different sub-group proportions are represented in the synthetic data as
compared to real data. Therefore representation biases are seen when the sub-
groups are missing, underrepresented, or overrepresented in the synthetic
data.

Figure 1 illustrates a simplified example wherein the degree of representation
in the synthetic data is measured using Partial Recall (?). The generated
distribution shows a substantial recall discrepancy between majority and
minority groups, notably expanding as the minority level increases.
Consequently, the synthetic data with minority samples exhibit both inadequate
quality and coverage problems, potentially resulting in incidental or spurious
correlations within the synthetic dataset.

Figure 1: An example of partial recall for groups with underrepresentation
(minor), overrepresentation (major), and adequate representation in data
generation.

In this paper, we address the problem of synthetic healthcare data fairness;
generate fair data from biased data (correlation biases), and promote
representation fairness in the target data.

Contributions. The problem definition, design, analysis, and experimental
assessment of the proposed framework are the main contributions of this
research. Particular contributions include:

  * •

Development of a principled GAN framework for synthetic healthcare data
generation with guaranteed fairness in the target in contrast to existing
techniques such as HealthGAN and MedGAN, where no fairness is ensured.

  * •

Problem definition considering how the DGP is affected by various biases from
the data (correlation bias) and how the generative model injects additional
biases (representation biases) in the training process.

  * •

A comprehensive experimental assessment of the proposed framework on health
database and comparison with state-of-the-art methods in terms of data utility
and fairness.

  * •

An analysis of the bias amplification (?) of the proposed approach and
comparison with the state-of-the-art.

  * •

An explainability analysis using SHAP (?) to validate the trustworthiness of
the proposed framework.

##  2 Related Works

We focus on the related literature in terms of i) synthetic data generation in
healthcare systems, ii) fair data generation, and (iii) representation issues
in GANs in contrast to fairness measures, which we define in Section 4.

GANs in Healthcare Systems. Synthea (?) simulates patient records from birth
to the present day using modules informed by clinicians and real-world
statistics. It claims to preserve utility by employing healthcare
practitioners and real statistics to construct rules that synthesize the data,
which assures privacy. HealthGAN (?) generates synthetic health data of
multivariate nature. The loss function is based on the Wasserstein distance
(?) and they used Synthetic Data Vault (SVD) (?) for categorical encoding.
Medical Image translation, MedGAN (?), is an end-to-end framework by merging
the adversarial network with non-adversarial losses. The discriminator network
is a pre-trained feature extractor, which penalizes the discrepancy. Among the
models mentioned above, HealthGAN provides promising results in terms of
accuracy and utility (?).

GANs in Fair Data Generation. The Fairness aware GAN (Fair GAN) (?) approach
trains a generator to produce fair representations by an additional
discriminator that knows the real distribution of the protected features in
the training data. DECAF (?) is designed to generate fair synthetic tabular
data with an assumption that the underlying causal structure is known. It uses
individual generators to generate features sequentially based on the causal
graph, while de-biasing is done at the inference time.

Improving Under-representation in GANs. GANs frequently experience mode
collapse and generate samples with low diversity because of the unstable
nature of the min-max game between a generator and a discriminator. Better
data coverage has been advocated using objective function-based methods (?),
(?) and structure-based methods (?), (?). Even while they work well to
increase data coverage overall, these methods don’t give minor modes any extra
attention. They frequently fail to recover minor modes when the minority ratio
for a given feature is meager. Techniques such as label smoothing have been
successfully applied to increase the performance of GANs when the scale of the
training data is limited (?). Another category of methods is sampling-based
methods (?) which give extra attention to minor modes and then promote these
modes by score-based sampling. Discriminator Rejection Sampling (DRS) (?) is
proposed to apply rejection sampling to filter the synthetic samples based on
density ratio estimation. Likewise, GOLD (?) reweighted fake samples to
improve the representation (or resemblance). Top-k training (?) modifies the
generator using only the top-k synthetic samples to promote representation.
Similar to DRS, Dia-GAN (?) proposed a discrepancy score based on the
empirical mean and variance over multiple epochs. We follow a similar approach
to tackle representation biases in data generation.

In addition, none of the above methods have successfully handled partially
unlabelled data. Medical data often contain missing labels which if not
treated properly can adversely affect a patient’s health.

In summary, we present Table 1 to compare the methods with different key areas
of interest. As far as we know, our Bt-GAN is the first GAN architecture that
tackles correlation biases, representation biases, and partially unlabeled
data in an end-to-end framework in the underlying DGP.

GAN | a | b | c | Method | Goal
---|---|---|---|---|---
HealthGAN | No | No | No | Wasserstein- | Synthetic
(?) |  |  |  | distance | health data
MedGAN | No | No | No | Progressive- | Synthetic
(?) |  |  |  | refinement | health data
FairGAN | Yes | No | No | Adversarial- | Fair synthetic
(?) |  |  |  | de-biasing | data
DECAF | Yes | No | No | Causal structure | Fair synthetic
(?) |  |  |  |  | tabular data
Bt-GAN(ours) | Yes | Yes | Yes | bias-transforming DGP + | Fair and representative
|  |  |  | score-based sampling | synthetic healthdata
Table 1: Comparison of related works with different key areas of interests:
(a) Correlation bias, (b) Representation bias, and (c) Partially unlabelled
data.

##  3 Unfairness in Data Generation Process

According to (?), the term bias-preserving in fairness literature means that
the status quo (or training dataset) is a baseline and the model tries to
reproduce the historic performances in the status quo, which only accounts for
direct discrimination. But, the bias-transforming metrics address indirect
discrimination by fixing the structural inequalities in the status quo. In the
following, we describe the causes of unfairness in DGP in detail and
illustrate how we approach the problem definition with the concepts studied in
(?).

###  3.1 Learning of Protected Attribute Information

Learning protected attribute information is one of the key elements of
unfairness in classification or prediction, as described in the literature
(?), (?), (?). This situation can be worse in synthetic data generation using
GANs as it amplifies the existing biases in the DGP as studied in (?).

Suppose we have an ideal biased dataset
$\mathcal{D}_{bias}=\\{\mathcal{\tilde{X}},{\mathcal{\tilde{Y}}},{\mathcal{\tilde{S}}}\\}$
where each label $\mathcal{\tilde{Y}}$ is correlated with each sensitive
attribute in equal intensity. Then for a well-trained fixed discriminator111we
assume that the data used to train is very large and the model has enough
capacity, the generative model, once converged, captures the dependencies from
$\mathcal{D}_{bias}$, which means the labels in the synthetic data are also
correlated with the sensitive attribute in equal intensity. Based on this and
motivated by (?), we define the following in the context of GANs and
correlation bias: Let $\mathcal{D}$ be the real dataset containing correlation
biases and $\hat{\mathcal{D}}$ be the synthetic data generated by the
underlying generator $G$ of a GAN.

Definition 1 - Bias-preserving DGP (Bp-DGP). A DGP is bias-preserving if and
only if the underlying generative model $G$, once optimized, learns to obtain
a transformation from a Multivariate Normal Distribution (MVD) to the real
data distribution by preserving the exact correlations from $\mathcal{D}$ and
replicate it in $\hat{\mathcal{D}}$ across sensitive groups.

Remark \- A Bp-DGP seeks to keep the historic performances from the real data
in the output of a target model when trained with synthetic data with
equivalent error rates for each group as shown in the real data.

###  3.2 Representation Bias: Failure of GANs in Capturing the Exact
Representations

The synthetic generation based on GANs fails to capture the exact sub-group
proportions from the real data as GANs try to match the distributions of real
data at the dataset level. We define the following in the context of GANs and
representation biases:

Definition 2 - Density-preserving DGP (Dp-DGP). A DGP is said to be density
preserving if and only if the underlying generative model $G$, once optimized
is learned to generate synthetic data $\hat{\mathcal{D}}$ in such a way that
the ratio of sub-groups between $\mathcal{D}$ and $\hat{\mathcal{D}}$ should
be the same.

Remark \- Due to the min-max objective of GAN optimization, the DGP can inject
additional biases in the form of representation biases due to the mode
collapse problem as studied in (?).

Correlation biases and representation biases are critical in synthetic data
generation based on GANS.

##  4 Desiderata

###  4.1 Fairness Definition

Formally, let $\mathcal{D}=\\{\mathcal{X},\mathcal{S},\mathcal{Y}\\}$ be a
dataset containing biases, where $X\in\mathcal{X}\subset\mathbb{R}^{d}$ is a
random variable of non-sensitive features, $S\in\mathcal{S}$ be sensitive
features and $Y\in\mathcal{Y}$, the labels. Also, let
$\mathcal{U}(\mathcal{S},\mathcal{Y})$ be a definition of algorithmic
fairness. We define the following algorithmic fairness measure to eliminate
indirect discrimination (?):

Definition 3 - (Statistical Parity (?)).  Suppose we have a function
$h:X\rightarrow Y^{\prime},Y^{\prime}=\\{0,1\\}$ for binary classification,
and assume $S$ splits $X$ into a majority set $\mathcal{M}$ and a minority set
$\mathcal{M}^{\prime}\left(X=\mathcal{M}\cup\mathcal{M}{}^{\prime}\right)$,
then the function $h$ satisfies statistical parity if $P[h(x)=1\mid
x\in\mathcal{M}]=P\left[h(x)=1\mid x\in\mathcal{M}^{\prime}\right]$, where $x$
denotes an instance of $X$ and $P[.]$ denotes the probability of an instance.

We assume the protected attribute is binary for notational convenience as this
can be extended to non-binary classification.

###  4.2 Mutual Information

Definition 4 - (Mutual Information (?)). Let $W_{1}$ and $W_{2}$ be two random
variables, the product of marginal distribution be $p_{W_{1}}$ $p_{W_{2}}$ and
the joint distribution be $p_{W_{1},W_{2}}$, then the mutual information
between $W_{1}$ and $W_{2}$ can be defined as:

$\begin{split}\@add@centering\centering I(W_{1};W_{2})&=H(W_{1})-H(W_{1}\mid
W_{2})=\\\
&\qquad\int_{W_{1}}\int_{W_{2}}p_{(W_{1},W_{2})}\log\frac{p_{(W_{1},W_{2})}}{p_{(W_{1})}p_{(W_{2})}}\,dW_{1}.\
dW_{2}\end{split}$ (1)

Unlike correlation coefficients (?) (which could only estimate the linear
dependence), mutual information is used to measure linear as well as non-
linear dependencies between two random variables.

Definition 5 - (Relationship between Statistical Parity and Zero Mutual
Information (?)) . Given a predicted outcome, $Y^{\prime}=\\{0,1\\}$, and a
protected attribute $S$, then,

$p\left[Y^{\prime}\mid S\right]=p\left[Y^{\prime}\right]\Leftrightarrow
p_{Y^{\prime},S}=p_{Y^{\prime}}p_{S}\Leftrightarrow
I\left(Y^{\prime};S\right)=0$ (2)

If the two random variables are independent of each other we get zero mutual
information.

###  4.3 Generative Adversarial Networks(GAN)

The Generative Adversarial Network (GAN) is a prominent member of the
generative models family, comprising two essential components: a generator and
a discriminator. The generator takes in random noise $z\sim\mathcal{N}(0,1)$
as input and endeavors to produce realistic data, represented as $x\sim
P_{\mathcal{D}}$. On the other hand, the discriminator is responsible for
distinguishing between the generated data and the original data. As time
progresses, the generator becomes more adept at deceiving the discriminator,
while the discriminator strives to differentiate between real and fake data.
This dynamic creates a zero-sum game, where both networks engage in a
continuous battle until they reach a state of equilibrium known as the Nash
equilibrium. To quantify the performance of the generator $(\mathrm{G})$ and
discriminator $(\mathrm{D})$, the following loss function is employed:

$\begin{split}\min_{\mathrm{G}}\max_{\mathrm{D}}&V(\mathrm{G},\mathrm{D})=E_{x\sim
P_{\mathcal{D}}}[log(\mathrm{D}(x))]+\\\ &E_{z\sim
p_{z}(z)}[log(1-\mathrm{D}(\mathrm{G}(z)))]\end{split}$ (3)

##  5 Synthetic Data Fairness

Synthetic data fairness means generating fair data from biased data such that
any downstream model trained on fair synthetic data will have fair predictions
in real data assuming that the underlying prediction model does not possess
any explicit biases.

To achieve synthetic data fairness considering correlation biases (Definition
1) and representation biases (Definition 2) in the DGP, we propose to define
Bias-transforming DGP (Bt-DGP) as:

Definition 6 - Bias-transforming DGP (Bt-DGP). Let $G$ be a generative model
and $\mathcal{U}(\mathcal{S},\mathcal{Y})$ be a definition of algorithmic
fairness (such as statistical parity in our case). A DGP is bias-transforming
if and only if it is Dp-DGP and the underlying generative model $G$ transforms
the existing biases in $\mathcal{D}$ in such a way that $\hat{\mathcal{D}}$ is
fair, as defined by $\mathcal{U}(\mathcal{S},\mathcal{Y})$ and the utility is
maintained with respect to any downstream tasks.

Figure 2: Architecture of Bt-GAN: the utilities of
$\mathrm{C}_{\varphi},\mathrm{G}_{\theta}$, $\mathrm{D}_{\zeta}$, and
$\mathrm{D}_{\phi}$ are shown. The symbols $A$ and $R$ denote accept and
reject respectively. The discriminator, $D_{\phi}$ accepts if it thinks it is
from the true data distribution denoted as $p$.

###  5.1 Problem Definition

The Synthetic Data Fairness Problem (SDFP) is to generate fair data
$\hat{\mathcal{D}}=\\{\hat{\mathcal{X}},\hat{\mathcal{S}},\hat{\mathcal{Y}\\}}$
from $\mathcal{D}$ through Bt-DGP.

Objective: To design a framework that accounts for both representation issues
and spurious correlations and to guarantee the fairness of any downstream
models trained on the synthetic data.

Approach: For tackling correlation biases we use and exploit Mutual
Information (MI) (?), (?), (?) as described in Section 4.2. For representation
biases, we adopt a score-based sampling as detailed in Section 6.

In the next section, we describe how we achieve Bt-DGP by proposing the Bt-GAN
framework.

##  6 Bias-transforming Generative Adversarial Networks

This section presents our Bias-transforming Generative Adversarial Network
(Bt-GAN) framework in detail.

The whole process of Bt-GAN can be divided into 3 stages:

  1. 1.

Pre-train and Diagnose \- the generator $G$ of a GAN learns to generate high-
quality samples from a large real-world dataset (biased and partially
labeled). During this process, we diagnose representation biases by recording
the corresponding sub-group densities from synthetic data.

  2. 2.

Bias-transform \- the pre-trained generator $G$ from stage 1 is fine-tuned to
generate fair distribution. This involves unlearning the sensitive
correlations using a fairness penalty and enforcing representation fairness
using score-based weighted sampling based on densities, computed from stage 1.
This transformation stage tackles both correlation bias from the data and the
representation bias from the GAN, thereby encouraging GAN to learn from the
under-represented regions of the data manifold.

  3. 3.

Discriminator Rejection Sampling (DRS) \- The score-weighted sampling in step
2 injects new biases towards under-represented regions. We correct it using
rejection sampling (?) using the discriminator trained in stage 1.

An overview of Bt-GAN architecture is given in Figure 2 and we describe all
the stages in detail as follows:

###  6.1 Pretrain and Diagnosis

Healthcare data often have missing values, especially partially unlabeled. So,
to completely capture the true data distribution from the partially unlabeled
data, we employ Triple GAN (?), a semi-supervised GAN framework involving 3
components: i) a classifier $C_{\varphi}$ for the conditional distribution,
$p_{\varphi}(y\mid x)$, ii) a Generator $G_{\theta}$ that characterizes
$p_{\theta}(x\mid y)\approx p(x\mid y)$, and iii) a Discriminator $D_{\phi}$
that predicts whether the data pair $(x,y)$ is from $p_{\theta}(x\mid y)$
(fake) or from $p(x\mid y)$ (real). After a sample $x$ is drawn, $C_{\varphi}$
produces $p_{\varphi}(x,y)=p(x)p_{\varphi}(y\mid x)$. Then, the joint
distribution produced by $G_{\theta}$ becomes,
$p_{\theta}(x,y)=p(y)p_{\theta}(x\mid y)$. Let $x$ be obtained by the latent
variable $z$, then, $x=$ $G_{\theta}(y,z),z\sim p_{z}(z)$, where $p_{z}(z)$
can be uniform or normal distribution.

The mini-max game for Triple GAN can be formulated as (?):

$\begin{split}L_{GCD}&=\min_{\varphi,\theta}\max_{\phi}V\left(C_{\varphi},G_{\theta},D_{\phi}\right)=\\\
&\qquad E_{(x,y)\sim p(x,y)}[\log D(x,y)]+\\\ &\lambda E_{(x,y)\sim
p_{\varphi}(x,y)}[\log(1-D(x,y))]+\\\ &(1-\lambda)E_{(x,y)\sim
p_{\theta}(x,y)}[\log(1-D(G(y,z),y))]+L_{CE},\end{split}$ (4)

where, $\lambda\in(0,1)$ is the balance factor that controls the significance
of classification and generation, we set $\lambda=0.5$ for the entire training
process. To make $p(x,y)=p_{\varphi}(x,y)=p_{\theta}(x,y)$, a cross entropy
loss $L_{CE}=E_{(x,y)\sim p(x,y)}\left[-\log p_{\varphi}(y\mid x)\right]$ has
been added (?).

During the training of the triple GAN, we record the densities for each
selected sub-groups. The densities are estimated by a measure called log
disparity of sub-groups (LDS) based on density ratios as follows:

Definition 7 - (Log Disparity of Sub-groups (LDS)).Let $f(x)$ be a membership
function for the binary definition of sub-groups $g_{j}\in\mathbb{G},1\leq
j\leq|\mathbb{G}|$, then $f(x)=1$, if $\mathrm{x}\in g_{j}$ and $f(x)=0$, if
$\mathrm{x}\notin g_{j}$. The log disparity of sub-groups, LDS(.) between
$p_{\mathcal{D}}$ and $p_{\hat{\mathcal{D}}}$, is defined as:

$\centering\operatorname{LDS}(x_{i})=\log\left(\frac{o\left(f(x_{i})=1\mid
x_{i}\in g_{j}\in p_{\hat{\mathcal{D}}}\right)}{o\left(f(x_{i})=1\mid x_{i}\in
g_{j}\in p_{\mathcal{D}}\right)}\right),\@add@centering$ (5)

where $o(f(x_{i})=1\mid x_{i}\in g_{j}\in p_{\hat{\mathcal{D}}})$ =
$P(f(x_{i})=1\mid x_{i}\in p_{\hat{\mathcal{D}}})/(1-(P(f(x_{i})=1\mid
x_{i}\in g_{j}\in p_{\hat{\mathcal{D}}})))$, and can similarly be calculated
for $o\left(f(x_{i})=1\mid x_{i}\in g_{j}\in p_{\mathcal{D}}\right)$. Log
disparity of sub-group can be computed for all the available sub-groups by
changing $f(x)$ to different $g_{j}$ in $\mathbb{G}$.

The discriminator $\mathrm{D}_{\phi}$ output can be used to estimate the
disparity of sub-groups, $LDS(.)$ between $\mathcal{D}$ and
$\hat{\mathcal{D}}$.

###  6.2 Bias Transformation

In the bias transformation step, the pre-trained GAN learns the fair
distribution underlying the chosen attributes. The fair distribution means
that the generated data should not contain both correlation biases and
representation biases.

####  6.2.1 Tackling Correlation Biases: Unlearning Sensitive Correlations
using Information-constrained DGP

GANs learn spurious correlations from data to converge to the true data
distribution. Let $W$ be the generated space. Specifically, we define learning
of protected attribute information as increasing $I(W;S)$ which is the mutual
information between $W$ and $S$ in the DGP. We exploit Definition 5 and
formulate a mutual information reduction problem between the generated space
$W$, which is a pair $\left\\{X^{\prime},Y^{\prime}\right\\}$, and the vector
encoded sensitive features, $S$ ($W$ and $S$ are random variables). So, we
propose the following information-constrained minimax objective function:

$\begin{split}\min_{\varphi,\theta}&\max_{\phi}V\left(C_{\varphi},G_{\theta},D_{\phi}\right)=E_{(x,y)\sim
p(x,y)}[\log D(x,y)]+\\\ &\lambda E_{(x,y)\sim
p_{\varphi}(x,y)}[\log(1-D(x,y))]+\\\ &(1-\lambda)E_{(x,y)\sim
p_{\theta}(x,y)}[\log(1-D(G(y,z),y))]+\\\ &L_{CE};\exists
I(W;S)=0,\end{split}$ (6)

The condition, $I(W;S)$ depends on the trainable parameter $\theta$. In this
setting, $S$ is a constant vector-encoded representation of sensitive features
selected by the data manager (or data owner).

For estimating the fairness penalty term $I(W;S)=0$, we use Mutual Information
Neural Estimation (MINE) (?), and the maximization can be handled by back-
propagation. The loss function $L_{I}$ for MINE is:

$L_{I}=E_{\hat{p}(W,S)}\left[T_{\eta}(W\mid S)\right]-\log
E_{\hat{p}_{w},\hat{p}_{s}}\left[e^{T_{\eta}(W\mid\hat{S})}\right],$ (7)

where $T_{\eta}$ is a statistical neural network. The loss term, $L_{I}$ can
provide an estimate of MI, once the parameter $\eta$ is maximized and it
depends on $\theta$. Also, $\hat{p}$ denotes the empirical estimation of
distribution. Therefore:

$L_{MI}=I[\widehat{W;S}]=\max_{\eta}L_{I}(\eta,\theta)$ (8)

The fairness penalty term helps the generator in learning a distribution $W$
from $Z$ with the constraint of not containing any malignant information
related to $S$.

So, the final loss, $L_{F}$ for Bt-GAN would be:

$L_{F}=\underbrace{\underbrace{L_{GCD}}_{Semi-supervised-
generation}+\underbrace{\alpha L_{MI}}_{MIde-biasing}}_{Fair-generation}$ (9)

The parameter $\alpha$ balances the MI reduction and the quality of
generation. We give an ablation study (Figure 3) on how it affects the
performance of the model by varying $\alpha$. The Triple GAN minimax objective
is optimized by iteratively modifying the generator, discriminator, and
classifier with respective losses.

Figure 3: Ablation study showing the effect of different values of $\alpha$ on
test and train accuracy. Note that, when $\alpha=1$, the effect of MI
reduction between $W$ and $S$ is large, but the accuracy also drops severely.
When $\alpha=0.5$, the model is performing comparatively better on the
mortality prediction task. Also, when $\alpha=0.0$, the MI reduction part in
equation 8 is inactive and thus is the fairness constraint. According to this,
we set $\alpha=0.5$ to balance the quality-fairness trade-off for the entire
process.

####  6.2.2 Improving Data Coverage using Density-preserving Sampling

Following (?), we propose a Log Disparity Sub-group Score (LDSS) (based on LDS
in equation 4), which measures how close the real and synthetic distributions
on sub-groups for training sample $x$ over $T$ iterations, as:

$LDSS(x_{i},T)=\frac{1}{T}\Sigma_{k\in T}LDS(x)_{k}$ (10)

The aim is to design a sampling probability for an instance $i$, based on
LDSS, and propagate it through SGD to design the batch size,
$\mathcal{D}_{\mathrm{B}}=x^{k}:x^{k}=x_{i},i\sim P_{LDSS}(i)$ for $k=$
$1,2,\ldots,B$. ie, each $x_{i}\in\mathcal{D}$ is sampled with a probability
of $P_{LDSS}(i)$.

Definition 8 - ($LDSS$-based Sampling Probability). For a training dataset
$\mathcal{D}$, we denote each instance as $x_{i}$ for notational convenience,
The sampling probability $P_{LDSS}(i)$ of the $i^{\text{th }}$ data instance
is calculated over a number of steps $T$ by:

$\
P_{LDSS}(i)=\frac{\operatorname{LDSS}\left(x_{i},T\right)}{\sum_{k=1}^{|\mathcal{D}|}\operatorname{LDSS}\left(x_{k},T\right)}$
(11)

Based on the LDSS, we define the following:

Definition 9 - ($LDSS$-based Density-preserving DGP). A DGP is LDSS-based Dp-
DGP, if and only if the LDSS of sub-groups (for each $g_{j}\in\mathbb{G}$ )
between ${p_{\mathcal{D}}}$ (real data distribution) and
$p_{\hat{\mathcal{D}}}$ (synthetic data distribution) is below an acceptable
threshold $\delta$.

We elaborate in Section 8.3 on how the threshold $\delta$ has been set to
various levels for evaluating the representation biases in synthetic health
data.

With reference to Definition 9, we re-iterate Definition 6 as:

Definition 10 - Bias-transforming DGP (Bt-DGP). Let $G$ be a generative model
and $\mathcal{U}(\mathcal{S},\mathcal{Y})$ be a definition of algorithmic
fairness (such as statistical parity in our case). A DGP is said to be bias-
transforming if and only if it is LDSS-based Dp-DGP and the underlying
generative model $G$ transforms the existing biases in $\mathcal{D}$ in such a
way that the $\hat{\mathcal{D}}$ is fair, evaluated by a definition of
algorithmic fairness, $\mathcal{U}(\mathcal{S},\mathcal{Y})$.

###  6.3 Discriminator Rejection Sampling

LDSS-based sampling creates biases as the generated distribution
$p_{\hat{\mathcal{D}}}$ differs from the real data distribution
$p_{\mathcal{D}}$. We employ rejection sampling (?) and use the discriminator
$D_{\phi}$ from stage 1 (as it knows the real distribution) with an acceptance
level $p_{\mathcal{D}}(x)/\operatorname{Lp}_{\hat{\mathcal{D}}}(x)$, for some
constant $L>0$. We use the same architecture for both discriminators (except
for the sigmoid activation). Also, to speed up the processing, the parameters
of $D_{\zeta}$ are instantiated with $D_{\phi}$.

##  7 Theorems and Proofs

Theorem 1:  An ideal data generation process222a process in which the
underlying generative model exactly converges to the true data distribution is
Bp-DGP.

Proof. For a well-trained fixed discriminator of a Triple GAN, $D_{\phi}$, the
global convergence of the optimization equation $L_{GCD}$ is achieved when
$p_{g}(x,y)=p(x,y)=p_{c}(x,y)$, where $p_{g}(x,y),p(x,y),$ and $p_{c}(x,y)$
respectively denote generator distribution, real data distribution, and
classifier distribution. Since the tree-player game is a hard constraint
compared to two-player optimization, it absolutely captures the real
dependence from data $\mathcal{D}$ even with missing labels. Note that we have
not changed the discriminator. Therefore the synthetic data
$\hat{\mathcal{D}}$ contains all the biases in $\mathcal{D}$, and hence it is
Bp-DGP.

Theorem 2: An ideal data generation process is Dp-DGP.

Proof. Let $G$ be the generator, $D$ be the discriminator, and $C$ be the
classifier of a Triple GAN. We assume enough capacity for $G$, $D$, and $C$.
Also, we assume that the Generator $G$ does not possess any mode collapse
under the ideal DGP and converges when $p_{g}(x,y)=p(x,y)=p_{c}(x,y)$ by
replicating all the sub-group densities from $\mathcal{D}$. Therefore, an
ideal DGP is Dp-DGP.

Theorem 3: An ideal DGP is both Bp-DGP and Dp-DGP.

Proof. It can be proved by Theorem 1 and Theorem 2.

Theorem 4: Given optimal discriminator $D_{C,G}^{*}(x,y)$, the global minimum
of the generator and classifier loss is achieved if and only if
$p(x,y)=p_{\phi}(x,y)=p_{\theta}(x,y)$.

We use the following Theorem to prove Theorem 4.

Theorem 4.1.  For any fixed $C$ and $G$, the optimal $D^{*}$ of the game
defined by the utility function $L_{GCD}$ is:

$D_{C,G}^{*}(x,y)=\frac{p(x,y)}{p(x,y)+p_{\gamma}(x,y)}$ (12)

where: $p_{\gamma}(x,y)=(1-\gamma)p_{g}(x,y)+\gamma p_{c}(x,y)$ is a mixture
distribution for $\gamma\in(0,1)$. We refer to Lemma 3.1, Lemma 3.2, and
Theorem 3.3 of (?) for the proof for Theorem 4.

Proof. Note that we have made zero changes to the Triple GAN discriminator and
therefore we can prove that both $\mathrm{C}$ and $\mathrm{G}$ will converge
to the true data distribution333we assume all the components have enough
capacity if in each iteration the discriminator has been trained to achieve
the optimum for a fixed $C$ and $G$ and $C$ and $G$ subsequently modified to
maximize the discriminator loss. Since the discriminator is trained optimum,
it penalizes for a very large deviation from real distribution even with the
addition of MI loss, thus finding a balance between accuracy and fairness (at
$\alpha=0.5$).

##  8 Experimental Results and Discussions

Dataset: We use MIMIC-III, a publicly available healthcare database containing
de-identified patient admissions between 2001 and 2012 (33798 unique patient
stays). We obtained permission to access MIMIC-III for research purposes after
completion of an online course (certification number 45456719). More details
regarding the cohort construction are given in Appendix.

Benchmarks: The methods we benchmark against are based on SOTA synthetic
health data generation schemes and fair data generation. We compare with
HealthGAN, as it is the best among the SOTA generative models in healthcare
data. Next, we validate the performance against FairGAN on how the fairness
measures have been improved in our settings. We could not find any references
regarding fair data generation in healthcare, addressing the concerns raised
in (?). We do not include DECAF (?) as it is based on the underlying causality
and finding causal inference knowledge on MIMIC-III is still a work-in-
progress (?), and we leave it for future study.

For the ablation study, we use two variants namely Bt-GAN-, which only
satisfies $\mathcal{U}(\mathcal{S},\mathcal{Y})$, and Bt-GAN which accounts
for both $\mathcal{U}(\mathcal{S},\mathcal{Y})$ and LDSS-based Dp-DGP.

Evaluation Metric: We evaluate our proposed model using the following
criteria:

  * •

Data Utility \- We use AUROC, AUPRC, F1 score, and accuracy for evaluating
data utility. We train classifiers (Linear Regression (LR) and Random Forest
(RF) on synthetic data for downstream tasks and validate the performance.
Moreover, we perform sample-level metrics analysis proposed in (?) together
with Jenson Shanon Divergence (JSD) (?) and discriminative score.

  * •

$\mathcal{U}(S,Y)$ Fairness \- This metric is used to measure the correlation
bias and we use the parity gap and AUROC gap. Moreover, we perform data and
model leakage (?) to compare the bias amplification in our settings (Details
in Appendix).

  * •

Representation Fairness \- It is compared and evaluated by $LDSS$ calculated
in every possible sub-groups. We use pie charts to plot the differences in
representations captured by different models.

Experimental Setup: The details of different neural networks are given in
Appendix.

###  8.1 Data Utility Analysis

We focus on four types of binary prediction tasks for analyzing the data
utility: (i) In-ICU mortality, (ii) In-hospital mortality, (iii) Length of
Stay (LOS) greater than 3 days, and (iv) Length of stay (LOS) greater than 7
days. We use LR and RF models for predictions. Our aim here is to compare the
quality and utility of the synthetic data generated by the proposed model, not
the classifier accuracy.

(a) AUROC_LR

(b) AUROC_RF

(c) AUPRC_LR

(d) AUPRC_RF

Figure 4: Data utility analysis | Real Data | HealthGAN | FairGAN | Bt-GAN
---|---|---|---|---
| accuracy. | F1 | accuracy. | F1 | accuracy. | F1 | accuracy. | F1
In-ICU mortality | 92.1 | 37.6 | 91.3 | 34.2 | 89.5 | 32.4 | 91.5 | 34.7
| 91.8 | 12.1 | 91.1 | 12.0 | 88.3 | 11.6 | 90.9 | 11.7
$LOS>3days$ | 71.2 | 59.9 | 69.4 | 58.3 | 67.1 | 56.3 | 68.1 | 57.3
| 72.6 | 59 | 67.2 | 59.1 | 66.4 | 57.9 | 68.9 | 57.6
In-hospital mortality | 90.1 | 39.6 | 89.1 | 37 | 85.4 | 32.8 | 89.6 | 39.9
| 89.3 | 17.9 | 88.3 | 15.8 | 86.3 | 14.3 | 90 | 18.1
$LOS>7days$ | 89.9 | 7.0 | 87.9 | 8.5 | 86.1 | 4.3 | 88.4 | 6.8
| 87.6 | 1.4 | 88.4 | 2.1 | 85.9 | 0.8 | 87.3 | 2.4
Table 2: Accuracy and F1 on various prediction tasks with real data as
reference point. For each task, the first row denotes the predictions by LR,
and the second row is the predictions by RF (higher is better for all the
values).

(a) HealthGAN

(b) Bt-GAN -

(c) Bt-GAN

Figure 5: Representation of sub-groups (The LDSS scores are given in brackets)
Figure 6: Comparison of log disparity values on the combination of attributes
such as mortality, age, and gender. The underrepresented (orange), over-
represented (blue), and adequately represented (green) demographic
combinations of the proposed models are compared with HealthGAN. The chart
areas are respectively divided as models, mortality, gender, and age.

(a) Real data

(b) HealthGAN

(c) FairGAN

(d) Bt-GAN

Figure 7: Importance of ethnicity among sub-groups by LR using SHAP (feature
ranks in the vertical axis)

Table 2 reports the accuracy and F1 score of the four downstream tasks. The
data generated by HealthGAN and Bt-GAN(ours) maintain the accuracy and F1
score for all the prediction tasks as that of real data, whereas FairGAN fails
to keep those. The performance of our model is commendable as it keeps almost
the same accuracy and F1 score as that of HealthGAN with added fairness.
Additionally, we compare utility in terms of AUROC and AUPRC using LR and RF.
Figure 4 shows 4 plots: (a) AUROC_LR, (b) AUROC _RF, (c) AUPRC_LR, and (d)
AUPRC_RF. The difference between HealthGAN and Bt-GAN is negligible in all
plots, whereas FairGAN achieves the worst performance. Finally, we provide
sample-level metrics analysis (?) to verify the quality, fidelity, diversity,
and generalization in Table 3. Our model is superior in the discriminative
score, $\beta$ recall, JSD, authenticity, and context FID compared to SOTA.

Model | Discriminative | JSD | $\alpha$ | $\beta$ | Authenticity | Context
---|---|---|---|---|---|---
| score |  | precision | recall |  | FID
|  ($\downarrow$) |  ($\downarrow$) |  ($\uparrow$) |  ($\uparrow$) |  ($\uparrow$) |  ($\downarrow$)
HealthGAN |  0.31$\pm$.002 |  0.032$\pm$.01 | 0.82$\pm$.002 |  0.52$\pm$.003 |  0.91$\pm$.001 |  0.89$\pm$.021
FairGAN |  0.46$\pm$.24 |  0.074$\pm$.05 |  0.56$\pm$0.53 |  0.21$\pm$0.12 |  0.62$\pm$.001 |  3.12$\pm$.32
Bt-GAN- |  0.35 $\pm$0.01 | 0.03$\pm$.21 |  0.70$\pm$.15 |  0.54$\pm$.04 |  0.88$\pm$.13 |  0.89$\pm$.13
Bt-GAN | 0.29 $\pm$0.01 |  0.031 $\pm$.001 |  0.810$\pm$.001 | 0.61 $\pm$0.032 | 0.92$\pm$.010 | 0.81$\pm$.12
Table 3: Quantitative analysis;$\uparrow$ indicates higher the better,
$\downarrow$ indicates lower the better, best results are bolded).

###  8.2 Fairness Analysis

MIMIC-III is composed of a set of sensitive features. As per Equal Credit
Opportunity Act [ECOA], gender, age, ethnicity, insurance type, and marital
status are considered sensitive information. To compare the fairness of the
proposed model with SOTA, we choose ethnicity on which we enforce
$\mathcal{U}(\mathcal{S},\mathcal{Y})$ fairness. Though there are no
restrictions in choosing the protected features in this study, enforcing
equality on age influences the patient’s health which may affect mortality or
length of stays in the ICU and cause unnecessary medical interventions. So, we
recommend the choice of protected features based on the context of
applications and suggestions from healthcare experts.

To compare $\mathcal{U}(\mathcal{S},\mathcal{Y})$ fairness in ethnicity sub-
groups, we choose two prediction tasks by LR: (i) In-hospital mortality, and
(ii) $LOS>7days$ on which we analyze the parity and AUROC gap between black
and white patients across various synthetic data.

Table 4 compares the parity as well as AUROC gaps in different GANs with real
data. The AUROC gap and the parity gap show that predictions on HealthGAN-
generated data are biased and amplified compared to real data. In all the
cases, a positive value represents a bias towards white patients and a
negative value represents a bias towards black patients. The data is fair when
the value is close to zero. FairGAN minimizes these biases by adversarial
debiasing. The gaps in Bt-GAN are almost close to zero for all the prediction
tasks considered. With this being said, the differences in parity, as well as
AUROC gaps between HealthGAN-generated data and Bt-GAN, are significant for
these predictions.

Metrics | Prediction | Real Data | HealthGAN | FairGAN | Bt-GAN
---|---|---|---|---|---
| In-hospital mortality |  0.043 $\pm$ 0.001 |  0.082 $\pm$ 0.002 |  0.021 $\pm$ 0.001 | 0.001 $\pm$ 0.001
AUROC | In-ICU mortality |  0.03 $\pm$ 0.007 |  0.15 $\pm$ 0.035 |  0.023 $\pm$ 0.064 | 0.012 $\pm$ 0.021
gap | LOS>3days |  -0.003 $\pm$ 0.002 |  -0.104 $\pm$ 0.001 |  -0.003 $\pm$ 0.001 | 0.000 $\pm$ 0.001
| LOS>7days |  -0.005 $\pm$ 0.002 |  -0.076 $\pm$ 0.002 |  -0.061 $\pm$ .001 | -0.013 $\pm$ 0.001
| In-hospital mortality |  -0.046 $\pm$ 0.018 |  -0.154 $\pm$ 0.010 |  -0.004 $\pm$ 0.014 | 0.000 $\pm$ 0.001
Parity | In-ICU mortality |  -0.031 $\pm$ 0.013 |  -0.331 $\pm$ 0.011 |  -0.005 $\pm$ 0.013 | 0.000 $\pm$ 0.000
gap | LOS>3days |  0.022 $\pm$ 0.012 |  0.224 $\pm$ 0.012 |  0.022 $\pm$ 0.002 | 0.000 $\pm$ 0.001
| LOS>7days |  -0.004 $\pm$ 0.002 |  -0.004 $\pm$ 0.002 | -0.002 $\pm$ 0.001 |  -0.003 $\pm$ 0.001
Table 4: Comparison of fairness gaps between white and black patients with
real data as reference (best results are in bold).

To further evaluate the data leakage and model leakage (?) of our proposed
model, we train an attacker (which is an ethnicity classifier) on the ground
truth labels (of various generated data) and the corresponding downstream
predictions (by LR) of the In-ICU mortality task. Note that the leakage of LR
on the real dataset is $0.60\pm 001$ with an F1 score of 92.3. This shows that
the underlying generative model amplifies these leakages as shown in Table 5.
Our proposed Bt-GAN models achieve superior performance in controlling these
leakages as the bias amplification factor $\Delta$, which is the difference
between the model and data leakage is less than zero (details in Appendix).
Note that adversarial de-biasing by FairGAN does not account for both
leakages, but is mitigated in Bt-GAN through MI-debiasing in the Bt-DGP.

Remark. The adversarial de-biasing-based generation process generates fair
data by fooling the discriminator. Our Bt-GAN method uses a module for
estimating the mutual information which does not compete with the generator.
That means we generate fair health data without fooling the estimator, but by
minimizing the information it estimates. The advantage of our method is that
we can train the estimator until convergence in every epoch which is not
possible in adversarial de-biasing methods.

Data | Data leakage | Model leakage
---|---|---
HealthGAN |  63.17$\pm$0.45 |  65.42$\pm$0.58
FairGAN |  60.81$\pm$0.32 |  62.97$\pm$0.31
Bt-GAN- |  50.45$\pm$0.10 |  49.16$\pm$0.14
Bt-GAN | 49.90$\pm$0.63 | 48.31$\pm$0.71
Table 5: Evaluation of data and model leakage.

###  8.3 Representation Fairness Analysis

It is essential for health sector research to incorporate demographic
information as part of clinical research. One such example could be to test
the effect of vaccines on different age groups, gender, or different physical
conditions. So, the real data should include appropriate proportions of all
the sub-groups to use for clinical research effectively. Thus, similar
distributions of sub-groups should be captured when it is being generated.

Our analysis of representation fairness can be performed in two ways: (i) on
sub-groups identified by gender, age, in-hospital mortality, and $LOS>3days$,
and (ii) on sub-groups defined by ICD-codes and ICU stay in the emergency care
unit. Note that, these sub-groups are selected not based on any clinical
research but to analyze how these proportions are being propagated into
synthetic data.

Similar to (?), we set the value of $\delta$ (Definition 9) to $\pm\log(0.9)$
(90 percent rule) as a threshold for over/underrepresented groups. Based on
this, we divide the range into 4 levels, such as missing $[-\propto$ to
$\log(.8)]$ (yellow), underrepresented $(\log(.8)$ to $\log(.9)]$ (orange),
adequately represented $(\log(.9)$ to $-\log(.9)]$ (green) and over-
represented $(-\log(.9)$ to $-\log(.8)]$ (blue).

Analysis of Sub-groups Identified by Gender, Age, In-hospital Mortality, and
$LOS>3days$.  We analyze the individual representation issues and the issues
caused by the intersection of these sub-groups in detail. Figure 5 (top row)
compares representation issues calculated by $LDSS$ for each of the sub-
groups. More representation biases (under/over) are caused by HealthGAN and
Bt-GAN--generated data. Specifically, more over-representation can be seen
towards the attributes defined by $age=^{\prime}81+^{\prime}$,
$mortality=^{\prime}Alive^{\prime}$, and $LOS>3days=^{\prime}No^{\prime}$. The
attributes $age=^{\prime}<=45^{\prime}$, $mortality=^{\prime}Died^{\prime}$,
and $LOS>3days=^{\prime}Yes^{\prime}$ are underrepresented. Suppose, if the
majority of the under-represented combinations belong to one particular class
and the over-represented in the other, the predictions by a downstream
classifier will be severely biased, even though the overall prediction
accuracy is acceptable.

Analysis of Sub-groups Identified by ICD Codes and ICU Stay in the Emergency
Care. We analyze the representation biases caused by ICD codes and various ICU
stays (CCU, CSRU, MICU, SICU, and TSICU) in Figure 5 (bottom row). This
analysis is very important in healthcare for setting up emergency care based
on appropriate disease characteristics (defined by ICD codes). Among the ICU
stays, CSRU and SICU are underrepresented in both the data generated by
HealthGAN and Bt-GAN-, whereas CCU is over-represented. The ICD codes
(ICD-0,3,4,5, and 9) are underrepresented in HealthGAN and Bt-GAN- (more
results are in Appendix). Note that, Bt-GAN captures all these sub-groups in
exact proportions in contrast to other schemes. Furthermore, we analyze the
representation issues caused by the cross-section of sub-groups in Figure 6.
Though the representations of gender are correctly captured by all the models,
their intersections with other demographics cause certain biases.

###  8.4 Attribute-based Local Explainability Analysis using SHAP

Motivated by (?), we analyze the group feature importance for samples in In-
hospital mortality prediction by LR and measure how attributed it is across
various sub-groups defined by the ethnicity group. We then rank the importance
among other features within SHAP exploratory analysis as shown in Figure 7. We
observe that the attribute ethnicity has a great impact (low ranks) on the
sub-groups White in real data, FairGAN, and HealthGAN resulting in biased
explanations (?). In Bt-GAN, the effect is more balanced and even reduced,
which means the feature ethnicity is less important to the model for all the
sub-groups. This local analysis is helpful in finding the disparities between
the sub-groups in situations where global explanations cannot reveal the
inequalities between the sub-groups.

##  9 Experiment on Fairness Benchmark Datasets

In this section, we evaluate the effectiveness of our model in producing fair
synthetic data on fairness benchmark datasets such as Adult
Income444https://archive.ics.uci.edu/ml/datasets/adult and
COMPAS555https://www.propublica.org/article/machine-bias-risk-assessments-in-
criminal-sentencing. Note that the missing labels in both of these datasets
are negligible. Therefore, semi-supervised learning has no impact on the
generation quality, and thus the terms associated with the classifier in Eq.
(4) have no relevance. As a result, Eq. (4) can be replaced by Eq. (3) for any
datasets with no (or negligible) missing labels.

###  9.1 Datasets

##### UCI Adult Dataset

This dataset is based on US census data (1994) and contains 48,842 rows with
attributes such as age, sex, occupation, and education level, and the target
variable indicates whether an individual has an income that exceeds $50K$ per
year or not. In our experiments, we consider the protected attribute to be sex
(S = “Sex”, Y = “Income”).

##### ProPublica Dataset from COMPAS Risk Assessment System

This dataset contains information about defendants from Broward County and
contains attributes about defendants such as their ethnicity, language,
marital status, sex, etc., and for each individual a score showing the
likelihood of recidivism (reoffending). In this experiment, we used a modified
version of the dataset. First, attributes such as FirstName, LastName,
MiddleName, CASE ID, and DateOfBirth are removed. Studies have shown that this
dataset is biased against African Americans. Therefore, ethnicity is chosen to
be the protected attribute for this study. Only African American and Caucasian
individuals are kept and the rest are dropped. The target variable in this
dataset is a risk decile score provided by the COMPAS system, showing the
likelihood of that individual to re-offend, which ranges from 1 to 10. The
final modified dataset contains 16,267 records with 16 features. To make the
target variable binary, a cut-off value of 5 is considered and individuals
with a decile score of less than 5 are considered “Low Chance”, while the rest
are considered “High Chance”. (S = “Ethnicity”, Y = “Recidivism Chance”).

##### Competing Methods

The methods we compare against are based on fair tabular data generation using
GANs. We compare against FairGAN and DECAF as these are state-of-the-art
methods in the tabular domain.

| Datasets | Precision | Recall | AUROC gap | Parity gap
---|---|---|---|---|---
| Adult |  0.902 $\pm$ 0.001 |  0.921 $\pm$ 0.002 | 0.198 ± 0.018 | 0.121 ± 0.024
Real Data | COMPAS |  0.903 $\pm$ 0.007 |  0.914 $\pm$ 0.007 |  0.239$\pm$ 0.002 |  0.258 $\pm$0.032
| Adult |  0.781 $\pm$ 0.018 |  0.881 $\pm$ 0.050 | 0.011 ± 0.002 | 0.001 ± 0.001
DECAF | COMPAS |  0.874 $\pm$ 0.010 |  0.886 $\pm$ 0.001 | 0.021 ± 0.010 | 0.003 ± 0.011
| Adult |  0.661 $\pm$ 0.140 |  0.679 $\pm$ 0.001 |  0.089$\pm$0.002 | 0.097 ± 0.018
FairGAN | COMPAS |  0.785 $\pm$ 0.010 |  0.832 $\pm$ 0.002 |  0.045$\pm$0.023 | 0.205 ± 0.055
| Adult |  0.898 $\pm$ 0.001 |  0.900 $\pm$ 0.020 | 0.002$\pm$ 0.010 | 0.001$\pm$ 0.020
Bt-GAN (ours) | COMPAS |  0.896 $\pm$ 0.001 |  0.899 $\pm$ 0.003 | 0.001 $\pm$ 0.002 | 0.002$\pm$ 0.001
Table 6: Data utility and fairness analysis on fairness benchmark datasets
with real data as the reference point (higher is better for all the values)

###  9.2 Results

We list the utility and fairness measures in Table 6. The precision and recall
of our proposed Bt-GAN are better than FairGAN and DECAF. The fairness
evaluated in terms of the AUROC gap and parity gap of Bt-GAN shows that the
bias in synthetic data has been reduced to a great extent compared to state-
of-the-art models. We achieve a good balance between utility and fairness
through a Bt-DGP.

##  10 Conclusion and Future Works

We proposed Bt-GAN, a semi-supervised Generative Adversarial Network for
synthetic healthcare data generation, which accounts for the spurious
correlations in the data and promotes representation fairness in the target in
an end-to-end framework. This is done by proposing a Bias-transforming
generation process, incorporating an information-constrained fairness penalty
to tackle correlation biases, a score-based weighted sampling to promote
representation fairness, and semi-supervised learning to capture true data
distribution under partially unlabeled data. Compared to the SOTA schemes, our
method is found to be more reliable to better balance the tradeoff between
accuracy and fairness with minimal bias amplification in a more generalized
and principled way.

Future Directions: This work is based on GANs and feature-specific constraints
such as mutual information. An interesting future direction could be to extend
this in the framework using diffusion models and causality. We leave further
experiments to extend our proposed methods from these perspectives as future
work.

Ethical and Societal Implications: The notion of fairness is domain and
context-dependent, which is more sensitive in healthcare. The proposed methods
could be coupled with tasks, where the goal is the progression of trustworthy
AI.

Acknowledgments

This work was funded by the Knut and Alice Wallenberg Foundation, the ELLIIT
Excellence Center at Linköping-Lund for Information Technology, and TAILOR -
an EU project with the aim to provide the scientific foundations for
Trustworthy AI in Europe. The computations were enabled by the Berzelius
resource provided by the Knut and Alice Wallenberg Foundation at the National
Supercomputer Centre.

##  A Quality and Fairness Definitions

##### Discriminative Score:

To calculate the score, we train a classifier(LR) to classify real or fake
with the original data and synthetic data respectively labeled as real and
fake and calculated the error.

##### Jensen-Shannon Divergence (JSD):

Here we used JSD to evaluate the quality of the synthetic data as compared to
the real data (?).

#####  $\alpha$-precision, $\beta$-recall, and Authenticity:

It is used to evaluate the fidelity, diversity, and generalization of the
synthetic data at the sample level as proposed in (?).

##### Context FID:

It is used to measure the difference in statistics between the real and
synthetic data with respect to downstream tasks (?).

##### Data Leakage:

It is used to evaluate how much information about the protected attributes can
be leaked through task-specific labels. To measure this, we train an attacker
$f$ and evaluate it on held-out data. The performance of the attacker, the
fraction of instances in $\mathcal{D}$ that leak information about $S_{i}$
through $Y_{i}$, yields an estimate of leakage (?):

$Leakage_{\mathcal{D}}=\frac{1}{|\mathcal{D}|}\sum_{\left(Y_{i},S_{i}\right)\in\mathcal{D}}{1}\left[f\left(Y_{i}\right)==S_{i}\right]$
(13)

##### Model Leakage:

To measure the degree a model, $M$ produces predictions,
$\hat{Y}_{i}=M\left(X_{i}\right)$, that leak information about the protected
variable $S_{i}$. We define model leakage as the percentage of examples in
$\mathcal{D}$ that leak information about $S_{i}$ through $\hat{Y}_{i}$. To
measure prediction leakage, we train a different attacker on $\hat{Y}_{i}$ to
extract information about $S_{i}$ (?) :

$Leakage_{M}=\frac{1}{|\mathcal{D}|}\sum_{\left(\hat{Y}_{i},S_{i}\right)\in\mathcal{D}}{1}\left[f\left(\hat{Y}_{i}\right)==S_{i}\right)]$
(14)

We measure fairness using two measures; the AUROC gap and the parity gap. The
AUROC gap is defined as the difference in AUROC between the selected sub-
groups for a particular healthcare task. The parity gap is the difference in
statistical parity between two sub-groups for a specified task. In Table 7, we
give definitions of the AUROC gap and Parity gap for sub_groups $SG_{i}$.

AUROC gap | $\operatorname{AUROC}\left(\mathrm{SG}_{1}\right)-\mathrm{AUROC}\left(\mathrm{SG}_{2}\right)$
---|---
Parity Gap | $\frac{TP_{1}+FP_{1}}{N_{1}}-\frac{TP_{2}+FP_{2}}{N_{2}}$
Table 7: Fairness definitions

##  B Extensions to Other Fairness Notions

Analogous to the connection between statistical parity and zero mutual
information, the fairness notions such as equalized opportunity and equalized
odds can be designed as a conditional mutual information optimization problem
as detailed in (?).

##  C Architecture Details

##### Implementation Details:

The training epochs use a mini-batch size of 1024. The learning rate is set to
.0001 with Adam optimizer. We carried out the experiments using PyTorch in
Intel Core i-9, 11th generation, with 128 GB RAM and GPU-2* NVIDIA RTX 2080TI
(11 GB). Also, the details architecture used in this study can be found on
Table 9.

(a) Real data

(b) HealthGAN

(c) Bt-GAN-

(d) Bt-GAN

Figure 8: Distribution of ICD-codes in ICU stays Figure 9: Heatmap
representation of ICD codes and ICU stays in real data, HealthGAN generated
data and Bt-GAN generated data

##### Dataset Preparation:

We extracted records from tables, PATIENTS, ADMISSIONS, ICU STAYS,
CHARTEVENTS, LABEVENTS, and OUTPUTEVENTS. Records are then validated using
HADM_ID and ICUSTAY_ID, resulting in a total of 33,798 patients with 42276 ICU
stays. Among them, we split 28728 patients, 35948 ICU stays for training, and
the remaining for testing. We excluded patients with missing HADM_ID and
ICUSTAY_ID.

##  D Additional Results

Additional results on the representation analysis are given in Figure 8 and
Figure 9.

Figure 10: The MI de-biasing accounts for proxy attribute

##  E Cohort Summary

Table 8 details the cohort summary defined by various demographic information.

Attributes | Groups | Percentage
---|---|---
| Black | 8%
| White | 71%
Ethnicity | Hispanic | 3%
| Other | 15%
| Asian | 2%
| Female | 43%
Gender | Male | 57%
| Self-pay | 1%
| Government | 3%
Insurance | Medicaid | 8%
| Private | 34%
| Medicare | 53%
| TSICU | 13%
| CCU | 15%
Emergency | SICU | 16%
| CSRU | 20%
| MICU | 35%
Total |  | 33798(100%)
Table 8: Cohort summary $\mathrm{C}_{\varphi}$ | $G_{\theta}$ |  $\mathrm{D}_{\phi}$ , $D_{\zeta}$ | $T_{\eta}$
---|---|---|---
z | No. of features | $(W,S)$ | $-T_{n}(\mathrm{~{}W})$
(normal distribution) | \- Dense layer of 64 | \- Dense layer of 64 | X - Dense layer of 8
\- Dense layer of 2 | \- leaky ReLU | \- leaky ReLU | \- ReLU
× no.of features -ReLU | \- Dense layer of 128 | \- Dense layer of 64 | \- Dense layer of 8
BN - Dense layer of 1.5 | \- leaky ReLU | \- leaky ReLU | \- ReLU
× no.of features- ReLU | \- Dense layer of 256 | \- Dense layer of 64 | \- Dense layer of 2
\- Dense layer of 1 | \- leaky ReLU | \- leaky ReLU | \- ReLU - Y
× no.of features- ReLU | \- Dense layer of 1 |  |
\- Gumble0.2 softmax(discrete) | \- leaky ReLU |  |
Table 9: Neural Networks for Generator, $G_{\theta}$, Discriminator,
$D_{\phi}$, $D_{\zeta}$, Classifier, $C_{\varphi}$, and MINE, $T_{\eta}$

##  F Impact of Proxy Attributes

We experimented further to study the impact of proxy attributes to check
whether these can introduce any biases. We added an extra feature,
$S\textunderscore{proxy}$ that is strongly correlated with ethnicity,
particularly for white and black sub-groups. For black patients,
$S\textunderscore{proxy}=1$ for 95 percent of all cases and
$S\textunderscore{proxy}=0$ for the remaining 5 percent. For white patients,
the above values are swapped. A correlation plot shows that there is a strong
correlation between ethnicity and $S\textunderscore{proxy}$ as well as the
combination of these to the mortality in the real health data. In our
synthetic data, the correlation between ’ethnicity’ and
$S\textunderscore{proxy}$ remains the same but the correlation of both of
these to the mortality is reduced to a great extent. So, it is evident that
the MI de-biasing accounts for proxy attributes as well (Figure 10).

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