# On Measuring Fairness in Generative Models

Christopher T. H. Teo
<EMAIL_ADDRESS>&Milad Abdollahzadeh
<EMAIL_ADDRESS>Cheung
<EMAIL_ADDRESS>Corresponding Author
Singapore University of Technology and Design (SUTD)

###### Abstract

Recently, there has been increased interest in fair generative models. In this
work, we conduct, for the first time, an in-depth study on fairness
measurement, a critical component in gauging progress on fair generative
models. We make three contributions. First, we conduct a study that reveals
that the existing fairness measurement framework has considerable measurement
errors, even when highly accurate sensitive attribute (SA) classifiers are
used. These findings cast doubts on previously reported fairness improvements.
Second, to address this issue, we propose CLassifier Error-Aware Measurement
(CLEAM), a new framework which uses a statistical model to account for
inaccuracies in SA classifiers. Our proposed CLEAM reduces measurement errors
significantly, e.g., 4.98%$\rightarrow$0.62% for StyleGAN2 w.r.t. Gender.
Additionally, CLEAM achieves this with minimal additional overhead. Third, we
utilize CLEAM to measure fairness in important text-to-image generator and
GANs, revealing considerable biases in these models that raise concerns about
their applications. Code and more resources: https://sutd-visual-computing-
group.github.io/CLEAM/.

##  1 Introduction

Figure 1:  ⓐ General framework for measuring fairness in generative models.
Generated samples with unknown ground-truth (GT) probability $\bm{p^{*}}$
w.r.t. sensitive attribute (SA) are fed into a SA classifier to obtain
$\bm{\hat{p}}$. Existing framework (Baseline) uses the classifier output
$\bm{\hat{p}}$ as estimation of $\bm{p^{*}}$. In contrast, our proposed CLEAM
includes an improved estimation that accounts for inaccuracies in the SA
classifier (see Alg. 1). \raisebox{-.9pt}b⃝ Our statistical model for fairness
measurement. This model accounts for inaccuracies in the SA classifier and is
the base of our proposed CLEAM (see Sec. 4.1). ⓒ Improvements with CLEAM.
CLEAM improves upon Baseline [1, 2] by reducing the relative error in
estimating the GT $p^{*}_{0}$ for SOTA GANs: StyleGAN2 [3] and StyleSwin [4],
and Stable Diffusion Model [5]. First row displays the Baseline and CLEAM
estimates for each GAN, using ResNet-18 as the SA classifier for Gender and
BlackHair. The Baseline incurs significant fairness measurement errors (e.g.
4.98%), even when utilizing a highly accurate ResNet-18 ($\approx$97%
accuracy). Meanwhile, CLEAM reduces the error significantly in all setups,
e.g. in the first panel, the error is reduced: 4.98% $\rightarrow$ 0.62%.
Similarly, in the second row, CLEAM reduces measurement error significantly in
the Stable Diffusion Model [5], using CLIP [6] as the SA classifier for
Gender, e.g. first panel: 9.14% $\rightarrow$ 0.05% (Detailed evaluation in
Tab. 1 and Tab. 2). Best viewed in color.

Fair generative models have been attracting significant attention recently [1,
2, 7, 8, 9, 10, 11, 12, 13]. In generative models [14, 15, 16, 17, 18],
fairness is commonly defined as equal generative quality [11] or equal
representation [1, 2, 7, 9, 12, 19, 20] w.r.t. some Sensitive Attributes (SA).
In this work, we focus on the more widely utilized definition – equal
representation. In this definition, as an example, a generative model is
regarded as fair w.r.t. Gender, if it generates Male and Female samples with
equal probability. This is an important research topic as such biases in
generative models could impact their application efficacy, e.g., by
introducing racial bias in face generation of suspects [21] or reducing
accuracy when supplementing data for disease diagnosis [22].

Fairness measurement for generative models. Recognizing the importance of fair
generative models, several methods have been proposed to mitigate biases in
generative models [1, 2, 7, 9, 12]. However, in our work, we focus mainly on
the accurate fairness measurement of deep generative models i.e. assessing and
quantifying the bias of generative models. This is a critical topic, as
accurate measurements are essential to reliably gauge the progress of bias
mitigation techniques. The general fairness measurement framework is shown in
Fig. 1 (See Sec. 2 for details). This framework is utilized in existing works
to assess their proposed fair generators. Central to the fairness measurement
framework is a SA classifier, which classifies the generated samples w.r.t. a
SA, in order to estimate the bias of the generator. For example, if eight out
of ten generated face images are classified as Male by the SA classifier, then
the generator is deemed biased at $0.8$ towards Male (further discussion in
Sec. 2). We follow previous works [1, 2, 12] and focus on binary SA due to
dataset limitations.

Research gap. In this paper, we study a critical research gap in fairness
measurement. Existing works assume that when SA classifiers are highly
accurate, measurement errors should be insignificant. As a result, the effect
of errors in SA classifiers has not been studied. However, our study reveals
that even with highly accurate SA classifiers, considerable fairness
measurement errors could still occur. This finding raises concerns about
potential errors in previous works’ results, which are measured using existing
framework. Note that the SA classifier is indispensable in fairness
measurement as it enables automated measurement of generated samples.

Our contributions. We make three contributions to fairness measurement for
generative models. As our first contribution, we analyze the accuracy of
fairness measurement on generated samples, which previous works [1, 2, 7, 9,
12] have been unable to carry out due to the unavailability of proper
datasets. We overcome this challenge by proposing new datasets of generated
samples with manual labeling w.r.t. various SAs. The datasets include
generated samples from Stable Diffusion Model (SDM) [5] —a popular text-to-
image generator— as well as two State-of-The-Art (SOTA) GANs (StyleGAN2 [3]
and StyleSwin [4]) w.r.t. different SAs. Our new datasets are then utilized in
our work to evaluate the accuracy of the existing fairness measurement
framework. Our results reveal that the accuracy of the existing fairness
measurement framework is not adequate, due to the lack of consideration for
the SA classifier inaccuracies. More importantly, we found that even in setups
where the accuracy of the SA classifier is high, the error in fairness
measurement could still be significant. Our finding raises concerns about the
accuracy of previous works’ results [1, 2, 12], especially since some of their
reported improvements are smaller than the margin of measurement errors that
we observe in our study when evaluated under the same setup; further
discussion in Sec. 3.

To address this issue, as our second (major) contribution, we propose
CLassifier Error-Aware Measurement (CLEAM), a new more accurate fairness
measurement framework based on our developed statistical model for SA
classification (further details on the statistical model in Sec. 4.1).
Specifically, CLEAM utilizes this statistical model to account for the
classifier’s inaccuracies during SA classification and outputs a more accurate
fairness measurement. We then evaluate the accuracy of CLEAM and validate its
improvement over existing fairness measurement framework. We further conduct a
series of different ablation studies to validate performance of CLEAM. We
remark that CLEAM is not a new fairness metric, but an improved fairness
measurement framework that could achieve better accuracy in bias estimation
when used with various fairness metrics for generative models.

As our third contribution, we apply CLEAM as an accurate framework to reliably
measure biases in popular generative models. Our study reveals that SOTA GANs
have considerable biases w.r.t. several SA. Furthermore, we observe an
intriguing property in Stable Diffusion Model: slight differences in
semantically similar prompts could result in markedly different biases for
SDM. These results prompt careful consideration on the implication of biases
in generative models. Our contributions are:

  * •

We conduct a study to reveal that even highly-accurate SA classifiers could
still incur significant fairness measurement errors when using existing
framework.

  * •

To enable evaluation of fairness measurement frameworks, we propose new
datasets based on generated samples from StyleGAN, StyleSwin and SDM, with
manual labeling w.r.t. SA.

  * •

We propose a statistically driven fairness measurement framework, CLEAM, which
accounts for the SA classifier inaccuracies to output a more accurate bias
estimate.

  * •

Using CLEAM, we reveal considerable biases in several important generative
models, prompting careful consideration when applying them to different
applications.

##  2 Fairness Measurement Framework

Fig.1(a) illustrates the fairness measurement framework for generative models
as in [1, 2, 7, 9, 12]. Assume that with some input e.g. noise vector for a
GAN or text prompt for SDM, a generative model synthesizes a sample
$\mathbf{x}$. Generally, as the generator does not label synthesized samples,
the ground truth (GT) class probability of these samples w.r.t. a SA (denoted
by $\bm{p^{*}}$) is unknown. Thus, an SA classifier $C_{\mathbf{u}}$ is
utilized to estimate $\bm{p^{*}}$. Specifically, for each sample
$\mathbf{x}\in\\{\mathbf{x}\\}$ , $C_{\mathbf{u}}(\mathbf{x})$ is the argmax
classification for the respective SA. In existing works, the expected value of
the SA classifier output over a batch of samples,
$\bm{\hat{p}}=\mathbb{E}_{\mathbf{x}}[C_{\mathbf{u}}(\mathbf{x})]$ (or the
average of $\bm{\hat{p}}$ over multiple batches of samples), is used as an
estimation of $\bm{p^{*}}$. This estimate may then be used in some fairness
metric $f$ to report the fairness value for the generator, e.g. fairness
discrepancy metric between $\bm{\hat{p}}$ and a uniform distribution
$\bm{\bar{p}}$ [1, 20] (see Supp A.3 for details on how to calculate $f$).
Note that the general assumption behind the existing framework is that with a
reasonably accurate SA classifier, $\bm{\hat{p}}$ could be an accurate
estimation of $\bm{p^{*}}$ [1, 9]. In the next section, we will present a
deeper analysis on the effects of an inaccurate SA classifier on fairness
measurement. Our findings suggest that there could be a large discrepancy
between $\bm{\hat{p}}$ and $\bm{p^{*}}$, even for highly accurate SA
classifiers, indicative of significant fairness measurement errors in the
current measurement framework.

One may argue that conditional GANs (cGANs) [23, 24] may be used to generate
samples conditioned on the SA, thereby eliminating the need for an SA
classifier. However, cGANs are not considered in previous works due to several
limitations. These include the limited availability of large labeled training
datasets, the unreliability of sample quality and labels [25], and the
exponentially increasing conditional terms, per SA. Similarly, for SDM,
Bianchi et al. [26] found that utilizing well-crafted prompts to mitigate
biases is ineffective due to the presence of existing biases in its training
dataset. Furthermore in Sec. 6, utilizing CLEAM, we will discuss that even
subtle prompt changes (while maintaining the semantics) result in drastically
different SA biases. See Supp G for further comparison between [26] and our
findings.

##  3 A Closer Look at Fairness Measurement

In this section, we take a closer look at the existing fairness measurement
framework. In particular, we examine its performance in estimating
$\bm{p^{*}}$ of the samples generated by SOTA GANs and SDM, a task previously
unstudied due to the lack of a labeled generated dataset. We do so by
designing an experiment to demonstrate these errors while evaluating biases in
popular image generators. Following previous works, our main focus is on
binary SA which takes values in $\\{0,1\\}$. Note that, we assume that the
accuracy of the SA classifier $C_{\mathbf{u}}$ is known and is characterized
by ${\bm{\alpha}}=\\{\alpha_{0},\alpha_{1}\\}$, where $\alpha_{i}$ is the
probability of correctly classifying label $i$. For example, for Gender
attribute, $\alpha_{0}$ and $\alpha_{1}$ are the probability of correctly
classifying Female, and Male classes, respectively. In practice,
$C_{\mathbf{u}}$ is trained on standard training procedures (more details in
the Supp F) and ${\bm{\alpha}}$ can be measured during the validation stage of
$C_{\mathbf{u}}$ and be considered a constant when the validation dataset is
large enough. Additionally, $\bm{p^{*}}$ can be assumed to be a constant
vector, given that the samples generated can be considered to come from an
infinite population, as theoretically there is no limit to the number of
samples from a generative model like GAN or SDM.

New dataset by labeling generators output. The major limitation of evaluating
the existing fairness measurement framework is the unavailability of
$\bm{p^{*}}$. To pave the way for an accurate evaluation, we create a new
dataset by manually labeling the samples generated by GANs and SDM. More
specifically, we utilize the official publicly released pre-trained StyleGAN2
[3] and StyleSwin [4] on CelebA-HQ [27] for sample generation. Then, we
randomly sample from these GANs and utilize Amazon Mechanical Turks to hand-
label the samples w.r.t. Gender and BlackHair, resulting in $\approx$9K
samples for each GAN; see Supp H for more details and examples. Next, we
follow a similar labeling process w.r.t. Gender, but with a SDM [5] pre-
trained on LAION-5B[28]. Here, we input prompts using best practices [26, 29,
30, 31], beginning with a scene description ("A photo with the face of"),
followed by four indefinite (gender-neutral) pronouns or nouns [32, 33] – {"an
individual", "a human being", "one person", "a person"} to collect $\approx$2k
high-quality samples. We refer to this new dataset as Generated Dataset
(GenData), which includes generated images from three models with
corresponding SA labels: GenData-StyleGAN2, GenData-StyleSwin, GenData-SDM. We
remark that these labeled datasets only provide a strong approximation of
$\bm{p^{*}}$ for each generator, however as the datasets are reasonably large,
we find this approximation sufficient and simply refer to it as the GT
$\bm{p^{*}}$. Then utilizing this GT $\bm{p^{*}}$, we compare it against the
estimated baseline ($\bm{\hat{p}}$). One interesting observation revealed by
GenData is that all three generators exhibit a considerable amount of bias
(see Tab.1 and 2); more detail in Sec. 6. Note that for a fair generator we
have $p^{*}_{0}=p^{*}_{1}=0.5$, and measuring the $p^{*}_{0}$ and $p^{*}_{1}$
is a good proxy for measuring fairness.

Experimental setup. Here, we follow Choi et al. [1] as the Baseline for
measuring fairness. In particular, to calculate each $\bm{\hat{p}}$ value for
a generator, a corresponding batch of $n=400$ samples is randomly drawn from
GenData and passed into $C_{\mathbf{u}}$ for SA classification. We repeat this
for $s=30$ batches and report the mean results denoted by
$\mu_{\texttt{Base}}$ and the 95% confidence interval denoted by
${\rho_{\texttt{Base}}}$. For a comprehensive analysis of the GANs, we repeat
the experiment using four different SA classifiers: Resnet-18, ResNet-34 [34],
MobileNetv2 [35], and VGG-16 [36]. Then, to evaluate the SDM, we utilize CLIP
[6] to explore the utilization of pre-trained models for zero-shot SA
classification; more details on the CLIP SA classifier in Supp. E. As CLIP
does not have a validation dataset, to measure ${\bm{\alpha}}$ for CLIP, we
utilize CelebA-HQ, a dataset with a similar domain to our application. We
found this to be a very accurate approximation; see Supp D.7 for validation
results. Note that for SDM, a separate $\bm{\hat{p}}$ is measured for each
text prompt as SDM’s output images are conditioned on the input text prompt.
As seen in Tab. 1 and 2, all classifiers demonstrate reasonably high average
accuracy $\in[84\%,98.7\%]$. Note that as we focus on binary SA (e.g.
Gender:{Male, Female}), both $\bm{p^{*}}$ and $\bm{\hat{p}}$ have two
components i.e. $\bm{p^{*}}=\\{p^{*}_{0},p^{*}_{1}\\}$, and
$\bm{\hat{p}}=\\{\hat{p}_{0},\hat{p}_{1}\\}$. After computing the
$\mu_{\texttt{Base}}$ and ${\rho_{\texttt{Base}}}$, we calculate normalized
$L_{1}$ point error $e_{\mu}$, and interval max error $e_{\rho}$ w.r.t. the
$p^{*}_{0}$ (GT) to evaluate the measurement accuracy of the baseline method:

$\displaystyle
e_{\mu_{\texttt{Base}}}=\tfrac{1}{p^{*}_{0}}|p^{*}_{0}-\mu_{\texttt{Base}}|\quad;\quad
e_{\rho_{\texttt{Base}}}=\tfrac{1}{p^{*}_{0}}\max\\{|\min(\rho_{\texttt{Base}})-p^{*}_{0}|,|\max(\rho_{\texttt{Base}})-p^{*}_{0}|\\}$
(1)

Based on our results in Tab. 1, for GANs, we observe that despite the use of
reasonably accurate SA classifiers, there are significant estimation errors in
the existing fairness measurement framework, i.e.
$e_{\mu_{\texttt{Base}}}$$\in[4.98\%,17.13\%]$. In particular, looking at the
SA classifier with the highest average accuracy of $\approx 97\%$ (ResNet-18
on Gender), we observe significant discrepancies between GT $p^{*}_{0}$ and
$\mu_{\texttt{Base}}$, with $e_{\mu_{\texttt{Base}}}=4.98\%$. These errors
generally worsen as accuracy marginally degrades, e.g. MobileNetv2 with
accuracy $\approx 96\%$ results in $e_{\mu_{\texttt{Base}}}=5.45\%$. These
considerably large errors contradict prior assumptions – that for a reasonably
accurate SA classifier, we can assume $e_{\mu_{\texttt{Base}}}$ to be fairly
negligible. Similarly, our results in Tab. 2 for the SDM, show large
$e_{\mu_{\texttt{Base}}}$$\in[1.49\%,9.14\%]$, even though the classifier is
very accurate. We discuss the reason for this in more detail in Sec. 5.1.

Overall, these results are concerning as they cast doubt on the accuracy of
prior reported results. For example, imp-weighting [1] which uses the same
ResNet-18 source code as our experiment, reports a 2.35% relative improvement
in fairness against its baseline w.r.t. Gender, which falls within the range
of our experiments smallest relative error, $e_{\mu_{\texttt{Base}}}$=4.98%.
Similarly, Teo et al. [2] and Um et al. [12] report a relative improvement in
fairness of 0.32% and 0.75%, compared to imp-weighting [1]. These findings
suggest that some prior results may be affected due to oversight of SA
classifier’s inaccuracies; see Supp. A.4 for more details on how to calculate
these measurements.

Remark: In this section, we provide the keystone for the evaluation of
measurement accuracy in the current framework by introducing a labeled dataset
based on generated samples. These evaluation results raise concerns about the
accuracy of existing framework as considerable error rates were observed even
when using accurate SA classifiers, an issue previously seen to be negligible.

##  4 Mitigating Error in Fairness Measurements

The previous section exposes the inaccuracies in the existing fairness
measurement framework. Following that, in this section, we first develop a
statistical model for the erroneous output of the SA classifier,
$\bm{\hat{p}}$, to help draw a more systematic relationship between the
inaccuracy of the SA classifier and error in fairness estimation. Then, with
this statistical model, we propose CLEAM – a new measurement framework that
reduces error in the measured $\bm{\hat{p}}$ by accounting for the SA
classifier inaccuracies to output a more accurate statistical approximation of
$\bm{p^{*}}$.

###  4.1 Proposed Statistical Model for Fairness Measurements

As shown in Fig.1(a), to measure the fairness of the generator, we feed $n$
generated samples to the SA classifier $C_{\mathbf{u}}$. The output of the SA
classifier ($\bm{\hat{p}}$) is in fact a random variable that aims to
approximate the $\bm{p^{*}}$. Here, we propose a statistical model to derive
the distribution of $\bm{\hat{p}}$.

As Fig.1(b) demonstrates in our running example of a binary SA, each generated
sample is from class 0 with probability $p^{*}_{0}$, or from class 1 with
probability $p^{*}_{1}$. Then, generated sample from class $i$ where
$i\in\\{0,1\\}$, will be classified correctly with the probability of
$\alpha_{i}$, and wrongly with the probability of
$\alpha^{\prime}_{i}=1-\alpha_{i}$. Thus, for each sample, there are four
mutually exclusive possible events denoted by ${\mathbf{c}}$, with the
corresponding probability vector ${\mathbf{p}}$:

$\displaystyle{\mathbf{c}}^{T}=\begin{bmatrix}c_{0|0}\quad c_{1|0}\quad
c_{1|1}\quad
c_{0|1}\end{bmatrix}\quad,\quad{\mathbf{p}}^{T}=\begin{bmatrix}p^{*}_{0}\alpha_{0}\quad
p^{*}_{0}\alpha_{0}^{\prime}\quad p^{*}_{1}\alpha_{1}\quad
p^{*}_{1}\alpha_{1}^{\prime}\end{bmatrix}$ (2)

where $c_{i|j}$ denotes the event of assigning label $i$ to a sample with GT
label $j$. Given that this process is performed independently for each of the
$n$ generated images, the probability of the counts for each output
${\mathbf{c}}^{T}$ in Eqn. 2 (denoted by $\mathbf{N}_{\mathbf{c}}$) can be
modeled by a multinomial distribution, i.e. $\mathbf{N}_{\mathbf{c}}\sim
Multi(n,{\mathbf{p}})$ [37, 38, 39]. Note that $\mathbf{N}_{\mathbf{c}}$
models the joint probability distribution of these outputs, i.e.
$\mathbf{N}_{\mathbf{c}}\sim\mathbb{P}(N_{c_{0|0}},N_{c_{1|0}},N_{c_{1|1}},N_{c_{0|1}})$
where, $N_{c_{i|j}}$ is the random variable of the count for event $c_{i|j}$
after classifying $n$ generated images. Since ${\mathbf{p}}$ is not near the
boundary of the parameter space, and as we utilize a large $n$, based on the
central limit theorem, $Multi(n,{\mathbf{p}})$ can be approximated by a
multivariate Gaussian distribution,
$\mathbf{N}_{\mathbf{c}}\sim\bm{\mathcal{N}}(\bm{\mu},\bm{\Sigma})$, with
$\bm{\mu}=n{\mathbf{p}}$ and $\bm{\Sigma}=n{\mathbf{M}}$ [40, 39], where
${\mathbf{M}}$ is defined as:

$\displaystyle{\mathbf{M}}=diag({\mathbf{p}})-{\mathbf{p}}{\mathbf{p}}^{T}$
(3)

$diag({\mathbf{p}})$ denotes a square diagonal matrix corresponding to vector
${\mathbf{p}}$ (see Supp A.1 for expanded form). The marginal distribution of
this multivariate Gaussian distribution gives us a univariate (one-
dimensional) Gaussian distribution for the count of each output
${\mathbf{c}}^{T}$ in Eqn. 2. For example, the distribution of the count for
event $c_{0|0}$, denoted by $N_{c_{0|0}}$, can be modeled as
$N_{c_{0|0}}\sim\mathcal{N}(\bm{\mu}_{1},\bm{\Sigma}_{11})$.

Lastly, we find the total percentage of data points labeled as class $i$ when
labeling $n$ generated images using the normalized sum of the related random
variables, i.e. $\hat{p}_{i}=\frac{1}{n}\sum_{j}N_{{c}_{i|j}}$. For our binary
example, $\hat{p}_{i}$ can be calculated by summing random variables with
Gaussian distribution, which results in another Gaussian distribution [41],
i.e. ,
$\hat{p}_{0}\sim\mathcal{N}(\tilde{\mu}_{\hat{p}_{0}},\tilde{\sigma}^{2}_{\hat{p}_{0}})$,
where:

$\displaystyle\begin{split}\tilde{\mu}_{\hat{p}_{0}}=&\tfrac{1}{n}(\bm{\mu}_{1}+\bm{\mu}_{4})=p^{*}_{0}\alpha_{0}+p^{*}_{1}\alpha_{1}^{\prime}\end{split}$
(4)
$\displaystyle\begin{split}\tilde{\sigma}^{2}_{\hat{p}_{0}}=&\tfrac{1}{n^{2}}(\bm{\Sigma}_{11}+\bm{\Sigma}_{44}+2\bm{\Sigma}_{14})=\tfrac{1}{n}[(p^{*}_{0}\alpha_{0}-(p^{*}_{0}\alpha_{0})^{2})+(p^{*}_{1}\alpha_{1}^{\prime}-(p^{*}_{1}\alpha_{1}^{\prime})^{2})]+\tfrac{2}{n}{p^{*}_{0}p^{*}_{1}\alpha_{0}\alpha_{1}^{\prime}}\end{split}$
(5)

Similarly
$\hat{p}_{1}\sim\mathcal{N}(\tilde{\mu}_{\hat{p}_{1}},\tilde{\sigma}^{2}_{\hat{p}_{1}})$
with $\tilde{\mu}_{\hat{p}_{1}}=(\bm{\mu}_{2}+\bm{\mu}_{3})/n$, and
$\tilde{\sigma}^{2}_{\hat{p}_{1}}=(\bm{\Sigma}_{22}+\bm{\Sigma}_{33}+2\bm{\Sigma}_{23})/n^{2}$
which is aligned with the fact that $\hat{p}_{1}=1-\hat{p}_{0}$.

Remark: In this section, considering the probability tree diagram in Fig.1(b),
we propose a joint distribution for the possible events of classification
($N_{c_{i|j}}$), and use it to compute the marginal distribution of each
event, and finally the distribution of the SA classifier outputs
($\hat{p}_{0}$, and $\hat{p}_{1}$). Note that considering Eqn. 4, 5, only with
a perfect classifier ($\alpha_{i}=1$, i.e. acc$=100\%$) the
$\tilde{\mu}_{\hat{p}_{0}}$ converges to $p^{*}_{0}$. However, training a
perfect SA classifier is not practical e.g. due to the lack of an appropriate
dataset and task hardness [42, 43]. As a result, in the following, we will
propose CLEAM which instead utilizes this statistical model to mitigate the
error of the SA classifier.

###  4.2 CLEAM for Accurate Fairness Measurement

In this section, we propose a new estimation method in fairness measurement
that considers the inaccuracy of the SA classifier. For this, we use the
statistical model, introduced in Sec 4.1, to compute a more accurate
estimation of $\bm{p^{*}}$. Specifically, we first propose a Point Estimate
(PE) by approximating the maximum likelihood value of $\bm{p^{*}}$. Then, we
use the confidence interval for the observed data ($\bm{\hat{p}}$) to propose
an Interval Estimate (IE) for $\bm{p^{*}}$.

Point Estimate (PE) for $\bm{p^{*}}$. Suppose that we have access to $s$
samples of $\bm{\hat{p}}$ denoted by
$\\{\bm{\hat{p}}^{1},\dots,\bm{\hat{p}}^{s}\\}$, i.e. SA classification
results on $s$ batches of generated data. We can then use the proposed
statistical model to approximate the $\bm{p^{*}}$. In the previous section, we
demonstrate that we can model $\hat{p}_{j}^{i}$ using a Gaussian distribution.
Considering this, first, we use the available samples to calculate sample-
based statistics including the mean and variance of the $\hat{p}_{j}$ samples:

$\displaystyle\begin{split}\ddot{\mu}_{\hat{p}_{j}}=&\textstyle\frac{1}{s}\textstyle\sum_{i=1}^{s}\hat{p}^{i}_{j}\end{split}$
(6)
$\displaystyle\begin{split}\ddot{\sigma}^{2}_{\hat{p}_{j}}=&{\textstyle\frac{1}{s-1}\textstyle\sum^{s}_{i=1}(\hat{p}^{i}_{j}-\ddot{\mu}_{\hat{p}_{j}})^{2}}\end{split}$
(7)

For a Gaussian distribution, the Maximum Likelihood Estimate (MLE) of the
population mean is its sample mean $\ddot{\mu}_{\hat{p}_{j}}$ [44]. Given that
$s$ is large enough (e.g. $s>30$), we can assume that
$\ddot{\mu}_{\hat{p}_{j}}$ is a good approximation of the population mean
[45], and equate it to the statistical population mean
$\tilde{\mu}_{\hat{p}_{j}}$ in Eqn. 4 (see Supp A.2 for derivation). With
that, we get the maximum likelihood approximation of $\bm{p^{*}}$, which we
call the CLEAM’s point estimate, $\mu_{\texttt{CLEAM}}$:

${\mu_{\texttt{CLEAM}}(p^{*}_{0})}=({\ddot{\mu}_{\hat{p}_{0}}-\alpha^{\prime}_{1}})/({\alpha_{0}-\alpha^{\prime}_{1}})\quad,\quad\mu_{\texttt{CLEAM}}(p^{*}_{1})=1-\mu_{\texttt{CLEAM}}(p^{*}_{0})$
(8)

Notice that $\mu_{\texttt{CLEAM}}$ accounts for the inaccuracy of the SA
classifier.

Require: accuracy of SA classifier, ${\bm{\alpha}}$.

1

2Compute SA classifier output
$\bm{\hat{p}}:\\{\bm{\hat{p}}^{1},\dots,\bm{\hat{p}}^{s}\\}$ for $s$ batches
of generated data.

3

4Compute sample mean $\ddot{\mu}_{\hat{p}}$ and sample variance
$\ddot{\sigma}^{2}_{\hat{p}}$ using (6) and (7).

5

6Use (8) to compute point estimate $\mu_{\texttt{CLEAM}}$.

7

8Use (10) to compute interval estimate $\rho_{\texttt{CLEAM}}$.

9

Algorithm 1 Computing point and interval estimates using CLEAM.

Interval Estimate (IE) for $\bm{p^{*}}$. In the previous part, we propose a PE
for $\bm{p^{*}}$ using the statistical model, and sample-based mean
$\ddot{\mu}_{\hat{p}_{0}}$. However, as we use only $s$ samples of
$\bm{\hat{p}}$, $\ddot{\mu}_{\hat{p}_{0}}$ may not capture the exact value of
the population mean. This adds some degree of inaccuracy into
$\mu_{\texttt{CLEAM}}$. In fact, in our framework, $\ddot{\mu}_{\hat{p}_{0}}$
equals $\tilde{\mu}_{\hat{p}_{0}}$ when $s\to\infty$. However, increasing each
unit of $s$ significantly increases the computational complexity, as each
$\bm{\hat{p}}$ requires $n$ generated samples. To address this, we recall that
$\hat{p}_{0}$ follows a Gaussian distribution and instead utilize frequentist
statistics [41] to propose a 95% confidence interval (CI) for $\bm{p^{*}}$. To
do this, first we derive the CI for $\tilde{\mu}_{\hat{p}_{0}}$:

$\ddot{\mu}_{\hat{p}_{0}}-1.96\tfrac{\ddot{\sigma}_{\hat{p}_{0}}}{\sqrt{s}}\leq\tilde{\mu}_{\hat{p}_{0}}\leq\ddot{\mu}_{\hat{p}_{0}}+1.96\tfrac{\ddot{\sigma}_{\hat{p}_{0}}}{\sqrt{s}}$
(9)

Then, applying Eqn.4 to Eqn.9 gives the lower and upper bounds of the
approximated 95% CI for $p^{*}_{0}$:

$\displaystyle\mathcal{L}(p^{*}_{0}),\mathcal{U}(p^{*}_{0})=(\ddot{\mu}_{\hat{p}_{0}}\mp
1.96(\ddot{\sigma}_{\hat{p}_{0}}/\sqrt{s})-\alpha_{1}^{\prime})/(\alpha_{0}-\alpha_{1}^{\prime})$
(10)

This gives us the interval estimate of CLEAM,
${\rho_{\texttt{CLEAM}}}=[\mathcal{L}(p^{*}_{0}),\mathcal{U}(p^{*}_{0})]$, a
range of values that we can be approximately 95% confident to contain
$p^{*}_{0}$. The range of possible values for $p^{*}_{1}$ can be simply
derived considering $p^{*}_{1}=1-p^{*}_{0}$. The overall procedure of CLEAM is
summarized in Alg. 1. Now, with the IE, we can provide statistical
significance to the reported fairness improvements.

Table 1:  Comparing the point estimates and interval estimates of Baseline
[1], Diversity [46] and our CLEAM in estimating $\bm{p^{*}}$ of StyleGAN2 [3]
and StyleSwin [4] with the proposed GenData datasets. We utilize SA
classifiers Resnet-18/34 (R18, R34)[34], MobileNetv2 (MN2)[35] and VGG-16
(V16)[36], with different accuracies ${\bm{\alpha}}$, to classify samples
w.r.t. attributes Gender and BlackHair. The $p^{*}_{0}$ value of each GAN
w.r.t. SA is determined by manually hand-labeling the generated data. We
repeat this for 5 experimental runs and report the mean error rate, per Eqn.
1. See Supp D.1 for the standard deviation of PE and IE.

|  |  | Point Estimate | Interval Estimate
---|---|---|---|---
| ${\bm{\alpha}}=\\{\alpha_{0},\alpha_{1}\\}$ | Avg. ${\bm{\alpha}}$ | Baseline | Diversity | CLEAM (Ours) | Baseline | Diversity | CLEAM (Ours)
|  |  | $\mu_{\texttt{Base}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{Div}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{CLEAM}}$ | $e_{\mu}(\downarrow)$ | $\rho_{\texttt{Base}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{Div}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{CLEAM}}$ | $e_{\rho}(\downarrow)$
(A) StyleGAN2
Gender with GT class probability $p^{*}_{0}$=0.642
R18 | {0.947, 0.983} | 0.97 | 0.610 | 4.98% | — | — | 0.638 | 0.62% | [0.602, 0.618] | 6.23% | — | — | [0.629, 0.646] | 2.02%
R34 | {0.932, 0.976]} | 0.95 | 0.596 | 7.17% | — | — | 0.634 | 1.25% | [0.589, 0.599] | 8.26% | — | — | [0.628, 0.638] | 2.18%
MN2 | {0.938, 0.975} | 0.96 | 0.607 | 5.45% | — | — | 0.637 |  0.78% | [0.602, 0.612] | 6.23% | — | — | [0.632, 0.643] | 1.56%
V16 | {0.801, 0.919} | 0.86 | 0.532 | 17.13% | 0.550 | 14.30% | 0.636 |  0.93% | [0.526, 0.538] | 18.06% | [0.536 , 0.564] | 16.51% | [0.628, 0.644] | 2.18%
Average Error: | 8.68% |  | 14.30% |  |  0.90% |  | 9.70% |  | 16.51% |  |  1.99%
BlackHair with GT class probability $p^{*}_{0}$=0.643
R18 | {0.869, 0.885} | 0.88 | 0.599 | 6.84% | — | — | 0.641 | 0.31% | [0.591, 0.607] | 8.08% | — | — | [0.631, 0.652] | 1.40%
R34 | {0.834, 0.916} | 0.88 | 0.566 | 11.98% | — | — | 0.644 |  0.16% | [0.561, 0.572] | 12.75% | — | — | [0.637, 0.651] | 1.24%
MN2 | {0.839, 0.881} | 0.86 | 0.579 | 9.95% | — | — | 0.639 |  0.62% | [0.574, 0.584] | 10.73% | — | — | [0.632, 0.647] | 1.71%
V16 | {0.851, 0.836} | 0.84 | 0.603 | 6.22% | 0.582 | 9.49% | 0.640 |  0.47% | [0.597, 0.608] | 7.15% | [0.568, 0.596] | 11.66% | [0.632, 0.648] | 1.71%
Average Error: | 8.75% |  | 9.49% |  |  0.39% |  | 9.68% |  | 11.66% |  |  1.52%
(B) StyleSwin
Gender with GT class probability $p^{*}_{0}$=0.656
R18 | {0.947, 0.983} | 0.97 | 0.620 | 5.49% | — | — | 0.648 | 1.22% | [0.612,0.629] | 6.70% | — | — | [0.639,0.658] | 2.59%
R34 | {0.932, 0.976} | 0.95 | 0.610 | 7.01% | — | — | 0.649 |  1.07% | [0.605,0.615] | 7.77% | — | — | [0.643,0.654] |  1.98%
MN2 | {0.938, 0.975} | 0.96 | 0.623 | 5.03% | — | — | 0.655 |  0.15% | [$0.618,0.629$] | $5.79\%$ | — | — | [0.649,0.661] |  1.07%
V16 | {0.801, 0.919} | 0.86 | 0.555 | 15.39% | 0.562 | 14.33% | 0.668 |  1.83% | [0.549,0.560] | 16.31% | [0.548,0.576] | 16.46% | [0.660,0.675] |  2.90%
Average Error: | 8.23% |  | 14.33% |  |  1.07% |  | 9.14% |  | 16.46% |  |  2.14%
BlackHair with GT class probability $p^{*}_{0}$=0.668
R18 | {0.869, 0.885} | 0.88 | 0.612 | 8.38% | — | — | 0.659 | 1.35% | [0.605,0.620] | 9.43% | — | — | [0.649,0.670] | 2.84%
R34 | {0.834, 0.916} | 0.88 | 0.581 | 13.02% | — | — | 0.662 |  0.90% | [0.576,0.586] | 13.77% | — | — | [0.656,0.669] |  1.80%
MN2 | {0.839, 0.881} | 0.86 | 0.596 | 10.78% | — | — | 0.659 |  1.35% | [0.591,0.600] | 11.50% | — | — | [0.652,0.666] |  2.40%
V16 | {0.851, 0.836} | 0.84 | 0.625 | 6.44% | 0.608 | 8.98% | 0.677 |  1.35% | [0.620,0.630] | 7.19% | [0.590,0.626] | 11.68% | [0.670,0.684] |  2.40%
Average Error: | 9.66% |  | 8.98% |  |  1.24% |  | 10.47% |  | 11.68% |  |  2.36%

Table 2: Comparing the point estimates and interval estimates of Baseline and
CLEAM in estimating the $\bm{p^{*}}$ of the Stable Diffusion Model [5] with
the GenData-SDM dataset. We use prompt input starting with "A photo with the
face of" and ending with synonymous (Gender neutral) prompts. We utilized CLIP
as the classifier for Gender, to obtain $\bm{\hat{p}}$.

|  | Point Estimate | Interval Estimate
---|---|---|---
Prompt | GT | Baseline | CLEAM (Ours) | Baseline | CLEAM (Ours)
| $p^{*}_{0}$ | $\mu_{\texttt{Base}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{CLEAM}}$ | $e_{\mu}(\downarrow)$ | $\rho_{\texttt{Base}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{CLEAM}}$ | $e_{\rho}(\downarrow)$
${\bm{\alpha}}$=[0.998,0.975], Avg. $\alpha$=0.987, CLIP –Gender
"A photo with the face of an individual" | 0.186 | 0.203 | 9.14% | 0.187 | 0.05% | [ 0.198 , 0.208 ] | 11.83% | [ 0.182 , 0.192 ] | 3.23%
"A photo with the face of a human being" | 0.262 | 0.277 | 5.73% | 0.263 | 0.38% | [ 0.270 , 0.285 ] | 8.78% | [ 0.255 , 0.271 ] | 3.44%
"A photo with the face of one person" | 0.226 | 0.241 | 6.63% | 0.230 | 1.77% | [ 0.232 , 0.251 ] | 11.06% | [ 0.220 , 0.239 ] | 5.75%
"A photo with the face of a person" | 0.548 | 0.556 | 1.49% | 0.548 | 0.00% | [ 0.545 , 0.566 ] | 3.28% | [ 0.537 , 0.558 ] | 2.01%
Average Error | 5.75% |  | 0.44% |  | 8.74% |  | 3.61%

Figure 2: LHS: Applying CLEAM to assess Gender bias in SDM[5] with CLIP [6].
Here, we utilize synonymous neutral prompts, prefixed with "A photo with the
face of __" as the input. We found that subtle changes to the prompts resulted
in significant changes in the bias. RHS: Illustrating the shift in Gender
bias. Using the same random seeds (per column), we generate 10 samples from
two prompts. Note how in some columns, Gender changes while retaining general
features e.g. pose.

##  5 Experiments

In this section, we first evaluate fairness measurement accuracy of CLEAM on
both GANs and SDM (Sec.5.1) with our proposed GenData dataset. Then we
evaluate CLEAM’s robustness through some ablation studies (Sec. 5.2). To the
best of our knowledge, there is no similar literature for improving fairness
measurements in generative models. Therefore, we compare CLEAM with the two
most related works: a) the Baseline used in previous works [1, 2, 7, 9, 12] b)
Diversity [46] which computes disparity within a dataset via an intra-dataset
pairwise similarity algorithm. We remark that, as discussed by Keswani et al.
[46] Diversity is model-specific using VGG-16 [36]; see Supp. D.2 for more
details. Finally, unless specified, we repeat the experiments with $s=30$
batches of images from the generators with batch size $n=400$. For a fair
comparison, all three algorithms use the exact same inputs. However, while
Baseline and Diversity ignore the SA classifier inaccuracies, CLEAM makes good
use of it to rectify the measurement error. As mentioned in Sec. 4.2, for
CLEAM, we utilize ${\bm{\alpha}}$ measured on real samples, which we found to
be a good approximation of the ${\bm{\alpha}}$ measured on generated samples
(see Supp. D.7 for results). We repeat each experiment 5 times and report the
mean value for each test point for both PE and IE. See Supp D.1 for the
standard deviation.

###  5.1 Evaluating CLEAM’s Performance

CLEAM for fairness measurement of SOTA GANs – StyleGAN2 and StyleSwin. For a
fair comparison, we first compute $s$ samples of $\bm{\hat{p}}$, one for each
batch of $n$ images. For Baseline, we use the mean $\bm{\hat{p}}$ value as the
PE (denoted by $\mu_{\texttt{Base}}$), and the $95\%$ confidence interval as
IE (${\rho_{\texttt{Base}}}$). With the same $s$ samples of $\bm{\hat{p}}$, we
apply Alg. 1 to obtain $\mu_{\texttt{CLEAM}}$ and ${\rho_{\texttt{CLEAM}}}$.
For Diversity, following the original source code [46], a controlled dataset
with fair representation is randomly selected from a held-out dataset of
CelebA-HQ [27]. Then, we use a VGG-16 [36] feature extractor and compute
Diversity, $\delta$. With $\delta$ we find $\hat{p}_{0}=(\delta+1)/2$ and
subsequently $\mu_{\texttt{Div}}$ and $\rho_{\texttt{Div}}$ from the mean and
$95\%$ CI (see Supp D.2 for more details on diversity). We then compute
$e_{\mu_{\texttt{CLEAM}}}$, $e_{\mu_{\texttt{Div}}}$,
$e_{\rho_{\texttt{CLEAM}}}$ and $e_{\rho_{\texttt{Div}}}$ with Eqn 1, by
replacing the Baseline estimates with CLEAM and Diversity.

As discussed, our results in Tab.1 show that the baseline experiences
significantly large errors of $4.98\%\leq e_{\mu_{\texttt{Base}}}\leq
17.13\%$, due to a lack of consideration for the inaccuracies of the SA
classifier. We note that this problem is prevalent throughout the different SA
classifier architectures, even with higher capacity classifiers e.g.
ResNet-34. Diversity, a method similarly unaware of the inaccuracies of the SA
classifier, presents a similar issue with $8.98\%\leq
e_{\mu_{\texttt{DiV}}}\leq 14.33\%$ In contrast, CLEAM dramatically reduces
the error for all classifier architectures. Specifically, CLEAM reduces the
average point estimate error from $e_{\mu_{\texttt{Base}}}\geq$ 8.23% to
$e_{\mu_{\texttt{CLEAM}}}\leq$ 1.24%, in both StyleGAN2 and StyleSwin. The IE
presents similar results, where in most cases $\rho_{\texttt{CLEAM}}$ bounds
the GT value of $\bm{p^{*}}$.

CLEAM for fairness measurement of SDM. We evaluate CLEAM in estimating the
bias of the SDM w.r.t. Gender, based on the synonymous (gender-neutral)
prompts introduced in Sec. 3. Recall that here we utilize CLIP as the zero-
shot SA classifier. Our results in Tab 2, as discussed, show that utilizing
the baseline results in considerable error ($1.49\%\leq
e_{\mu_{\texttt{Base}}}\leq 9.14\%$) for all prompts, even though the SA
classifier’s average accuracy was high, $\approx 98.7\%$ (visual results in
Fig.2). A closer look at the theoretical model’s Eqn. 4 reveals that this is
due to the larger inaccuracies observed in the biased class
($\alpha_{1}^{\prime}$) coupled with the large bias seen in $p^{*}_{1}$, which
results in $\mu_{\texttt{Base}}$ deviating from $p^{*}_{0}$. In contrast,
CLEAM accounts for these inaccuracies and significantly minimizes the error to
$e_{\mu_{\texttt{CLEAM}}}\leq 1.77\%$. Moreover, CLEAM’s IE is able to
consistently bound the GT value of $p^{*}_{0}$.

###  5.2 Ablation Studies and Analysis

Here, we perform the ablation studies and compare CLEAM with classifier
correction methods. We remark that detailed results of these experiments are
provided in the Supp due to space limitations.

CLEAM for measuring varying degrees of bias. As we cannot control the bias in
trained generative models, to simulate different degrees of bias, we evaluate
CLEAM with a pseudo-generator. Our results show that CLEAM is effective at
different biases ($p^{*}_{0}\in$ [0.5,0.9]) reducing the average error from
2.80% $\leq e_{\mu_{\texttt{Base}}}\leq$ 6.93% to
$e_{\mu_{\texttt{CLEAM}}}\leq$ 0.75% on CelebA [47] w.r.t. {Gender,BlackHair},
and AFHQ [48] w.r.t. Cat/Dog. See Supp D.3 and D.4 for full experimental
results.

CLEAM vs Classifier Correction Methods [49]. CLEAM generally accounts for the
classifier’s inaccuracies, without targeting any particular cause of
inaccuracies, for the purpose of rectifying the fairness measurements. This
objective is unlike traditional classifier correction methods as it does not
aim to improve the actual classifier’s accuracy. However, considering that
classifier correction methods may improve the fairness measurements by
directly rectifying the classifier inaccuracies, we compare its performance
against CLEAM. As an example, we utilize the Black Box Shift Estimation /
Correction (BBSE / BBSC) [49] which considers the label shift problem and aims
to correct the classifier output by detecting the distribution shift. Our
results, based on Sec. 5.1 setup, show that while BBSE does improve on the
fairness measurements of the baseline i.e. 4.20% $\leq
e_{\mu_{\texttt{BBSE}}}\leq$ 3.38%, these results are far inferior to CLEAM’s
results seen in Tab. 1. In contrast, BBSC demonstrates no improvements in
fairness measurements. See Supp D.8 for full experimental results. We
postulate that this is likely due to the strong assumption of label shift made
by both methods.

Effect of batch-size. Utilizing experimental setup in Sec. 5.1 for batch size
$n\in$[100,600], our results in Fig. 9 show that $n$=400 is an ideal batch
size, balancing computational cost and measurement accuracy. See Supp F for
full experimental details and results.

Figure 3:  Comparing the point error $e_{\mu}$ for Baseline and CLEAM when
evaluating the bias of GenData-CelebA with ResNet-18, while varying sample
size, $n$.

##  6 Applying CLEAM: Bias in Current SOTA Generative Models

In this section, we leverage the improved reliability of CLEAM to study biases
in the popular generative models. Firstly, with the rise in popularity of
text-to-image generators [5, 50, 51, 52], we revisit our results when passing
different prompts, with synonymous neutral meanings to an SDM, and take a
closer look at how subtle prompt changes can impact bias w.r.t. Gender.
Furthermore, we further investigate if similar results would occur in other
SA, Smiling. Secondly, with the shift in popularity from convolution to
transformer-based architectures [53, 54, 55], due to its better sample
quality, we determine whether the learned bias would also change. For this, we
compare StylesSwin (transformer) and StyleGAN2 (convolution), which are both
based on the same architecture backbone.

Our results, on SDM, demonstrate that the use of different synonymous neutral
prompts [32, 33] results in different degrees of bias w.r.t. both Gender and
Smiling attributes. For example in Fig. 2, a semantically insignificant prompt
change from "one person" to "a person" results in a significant shift in
Gender bias. Then, in Fig. 4(a), we observe that while the SDM w.r.t. our
prompts appear to be heavily biased to not-Smiling, having "person" in the
prompt appears to significantly reduce this bias. This suggests that for SDM,
even semantically similar neutral prompts [32, 33] could result in different
degrees of bias, thereby demonstrating certain instability in SDM. Next, our
results in Fig. 4(b) compare the bias in StyleGAN2, StylesSwin, and the
training CelebA-HQ dataset over an extended number of SAs. Overall, we found
that while StyleSwin produces better quality samples [4], the same biases
still remain statistically unchanged between the two architectures i.e. their
IE overlap. Interestingly, our results also found that both the GANs were less
biased than the training dataset itself.

(a) Evaluating bias of SDM [5] w.r.t. SA Smiling using CLEAM. LHS: Bias value
with different text prompts as input to SDM. CLEAM identifies that having
“person” in the prompt results in more Smiling samples generated. RHS: Visual
comparison of the samples generated for two different but semantically similar
text prompts. Samples in the same column are generated using the same random
seed. Notice how the SA (smiling) of the samples changes with a slight
difference in the prompts.

(b) CLEAM on StyleGAN2 [3] and StyleSwin [4], both pre-trained on CelebA-HQ
but are based on different architecture variants. We utilize a ResNet-18 and
compare the bias w.r.t. various SAs.

Figure 4: Applying CLEAM to further assess the bias in popular generative
models.

##  7 Discussion

Conclusion. In this work, we address the limitations of the existing fairness
measurement framework. Since generated samples are typically unlabeled, we
first introduce a new labeled dataset based on three state-of-the-art
generative models for our studies. Our findings suggest that the existing
framework, which ignores classification inaccuracies, suffers from significant
measurement errors, even when the SA classifier is very accurate. To rectify
this, we propose CLEAM, which considers these inaccuracies in its statistical
model and outputs a more accurate fairness measurement. Overall, CLEAM
demonstrates improved accuracy over extensive experimentation, including both
real generators and controlled setups. Moreover, by applying CLEAM to popular
generative models, we uncover significant biases that raise efficacy concerns
about these models’ real-world application.

Broader Impact. Given that generative models are becoming more widely
integrated into our everyday society e.g. text-to-image generation, it is
important that we have reliable means to measure fairness in generative
models, thereby allowing us to prevent these biases from proliferating into
new technologies. CLEAM provides a step in this direction by allowing for more
accurate evaluation. We remark that our work does not introduce any social
harm but instead improves on the already existing measurement framework.

Limitations. Despite the effectiveness of the proposed method along various
generative models, our work addresses only one facet of the problems in the
existing fairness measurement and there is still room for further improvement.
For instance, it may be beneficial to consider SA to be non-binary e.g. when
hair color is not necessary fully black (grey). Additionally, existing
datasets used to train classifiers are commonly human-annotated, which may
itself contain certain notions of bias. See Supp. I for further discussion.

## Acknowledgements

This research is supported by the National Research Foundation, Singapore
under its AI Singapore Programmes (AISG Award No.: AISG2-TC-2022-007) and SUTD
project PIE-SGP-AI-2018-01. This research work is also supported by the Agency
for Science, Technology and Research (A*STAR) under its MTC Programmatic Funds
(Grant No. M23L7b0021). This material is based on the research/work support in
part by the Changi General Hospital and Singapore University of Technology and
Design, under the HealthTech Innovation Fund (HTIF Award No. CGH-
SUTD-2021-004). We thank anonymous reviewers for their insightful feedback and
discussion.

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## Supplementary Material

This supplementary provides additional experiments as well as details that are
required to reproduce our results. These were not included in the main paper
due to space limitations. The supplementary is arranged as follows:

  * •

Section A: Details on Modelling

    * –

Section A.1 Details of Theoretical Modelling

    * –

Section A.2 Additional Details on CLEAM Algorithm

    * –

Section A.3 Details on Fairness Metric

    * –

Section A.4 Details of Significance of the Baseline Errors

  * •

Section B: Deeper Analysis on Error in Fairness Measurement

  * •

Section C: Validating Statistical Model for Classifier Output

    * –

Section C.1 Validation of Sample-Based Estimate vs Model-Based Estimate

    * –

Section C.2 Goodness-of-Fit Test: $\bm{\hat{p}}$ from the Real GANs with Our
Theoretical Model

  * •

Section D: Additional Experimental Results

    * –

Section D.1 Experimental Results with Standard Deviation

    * –

Section D.2 Experimental Setup for Diversity

    * –

Section D.3 Measuring Varying Degrees of Bias (Gender and BlackHair)

    * –

Section D.4 Measuring Varying Degrees of Bias with Additional Sensitive
Attributes (Young and Attractive)

    * –

Section D.5 Measuring Varying Degrees of Bias with Additional Sensitive
Attribute Classifiers (MobileNetV2)

    * –

Section D.6 Measuring SOTA GANs and Diffusion Models with Additional
Classifier (CLIP)

    * –

Section D.7 Comparing Classifiers Accuracy on Validation Dataset vs Generated
Dataset

    * –

Section D.8 Comparing CLEAM with Classifier Correction Techniques (BBSE/BBSC)

    * –

Section D.9 Applying CLEAM to Re-evaluate Bias Mitigation Algorithms

  * •

Section E: Details on Applying CLIP as a SA Classifier

  * •

Section F: Ablation Study: Details of Hyper-Parameter Settings and Selection

  * •

Section G: Related work

  * •

Section H: Details of the GenData: A New Dataset of Labeled Generated Images

  * •

Section I: Limitations and Considerations

##  Appendix A Details on Modelling

###  A.1 Details of Theoretical Modelling

In Sec 4.1 of main paper, we have proposed a statistical model for the
sensitive attribute classifier output which is then used in CLEAM to rectify
current measurement method. In this section, we give more details of this
model which is not included in the main paper due to lack of space.

Recall that in main paper, we mentioned that there are four possible mutually
exclusive outputs ${\mathbf{c}}$ for each sample with corresponding
probability vector ${\mathbf{p}}$:

${\mathbf{c}}=\begin{bmatrix}c_{0|0}\\\ c_{1|0}\\\ c_{1|1}\\\ c_{0|1}\\\
\end{bmatrix};\quad{\mathbf{p}}=\begin{bmatrix}p^{*}_{0}\alpha_{0}\\\
p^{*}_{0}\alpha_{0}^{\prime}\\\ p^{*}_{1}\alpha_{1}\\\
p^{*}_{1}\alpha_{1}^{\prime}\\\ \end{bmatrix}$

where $c_{i|j}$ denotes the event of assigning label $i$ to a sample with GT
label $j$. Then, we mentioned that the count for each ouptut can be modeled as
a multinomial distribution, $\mathbf{N}_{{\mathbf{c}}}\sim
Multi(n,{\mathbf{p}})$. Note that
$\mathbf{N}_{\mathbf{c}}=[N_{c_{0|0}},N_{c_{1|0}},N_{c_{1|1}},N_{c_{0|1}}]^{T}$
is the random vector of counts for individual outputs of ${\mathbf{c}}$.
$N_{c_{i|j}}$ is the random variable of the count for event $c_{i|j}$ after
classifying $n$ generated images. First, we consider following assumptions:

  1. 1.

Classifiers are reasonably accurate. We state that, given the advancement in
classifiers architecture, and the assumption that the sensitive attribute
classifier is trained with proper training procedures, it is a reasonable
assumption that it achieves reasonable accuracy and hence, $\alpha_{0}\neq 0$
and $\alpha_{1}\neq 0$. Similarly, we assume that it is highly unlikely to
have a perfect classifier and as such $\alpha^{\prime}_{0}=1-\alpha_{0}\neq 0$
and $\alpha^{\prime}_{1}=1-\alpha_{1}\neq 0$.

  2. 2.

Generators are not completely biased. Given that a generator is trained on a
reliable dataset with the availability of all classes of a given sensitive
attribute, coupled with the advancement in generator’s architecture, it is a
fair assumption that the generator would learn some representation of each
class in the sensitive attribute and not be completely biased, as such
$p^{*}_{0}\neq 0$ and $p^{*}_{1}\neq 0$.

Based on this assumptions, ${\mathbf{p}}$ is not near the boundary of the
parameter space, and we can conclude that $0<{\mathbf{p}}<1$. Therefore, we
can approximate the multinomial distribution as a Gaussian,
$\mathbf{N}_{\mathbf{c}}\sim\bm{\mathcal{N}}(\bm{\mu},\bm{\Sigma})$, with
$\bm{\mu}=n{\mathbf{p}}$ and $\bm{\Sigma}=n{\mathbf{M}}$ [56], where

$\bm{M}=\begin{bmatrix}p^{*}_{0}\alpha_{0}&0&0&0\\\
0&p^{*}_{0}\alpha_{0}^{\prime}&0&0\\\ 0&0&p^{*}_{1}\alpha_{1}&0\\\
0&0&0&p^{*}_{1}\alpha_{1}^{\prime}\\\
\end{bmatrix}-\begin{bmatrix}(p^{*}_{0}\alpha_{0})^{2}&{(p^{*}_{0})}^{2}\alpha_{0}\alpha_{0}^{\prime}&p^{*}_{0}p^{*}_{1}\alpha_{0}\alpha_{1}&p^{*}_{0}p^{*}_{1}\alpha_{0}\alpha_{1}^{\prime}\\\
{(p^{*}_{0}})^{2}\alpha_{0}\alpha_{0}^{\prime}&(p^{*}_{0}\alpha_{0}^{\prime})^{2}&p^{*}_{0}p^{*}_{1}\alpha_{0}^{\prime}\alpha_{1}&p^{*}_{0}p^{*}_{1}\alpha_{0}^{\prime}\alpha_{1}^{\prime}\\\
p^{*}_{0}p^{*}_{1}\alpha_{0}\alpha_{1}&p^{*}_{0}p^{*}_{1}\alpha_{0}^{\prime}\alpha_{1}&(p^{*}_{1}\alpha_{1})^{2}&(p^{*}_{1})^{2}\alpha_{1}\alpha_{1}^{\prime}\\\
p^{*}_{0}p^{*}_{1}\alpha_{0}\alpha_{1}^{\prime}&p^{*}_{0}p^{*}_{1}\alpha_{0}^{\prime}\alpha_{1}^{\prime}&(p^{*}_{1})^{2}\alpha_{1}\alpha_{1}^{\prime}&(p^{*}_{1}\alpha_{1}^{\prime})^{2}\\\
\end{bmatrix}$

and therefore:

$\bm{\mu}=\begin{bmatrix}\bm{\mu}_{1}\\\ \bm{\mu}_{2}\\\ \bm{\mu}_{3}\\\
\bm{\mu}_{4}\\\ \end{bmatrix}=n\begin{bmatrix}p^{*}_{0}\alpha_{0}\\\
p^{*}_{0}\alpha_{0}^{\prime}\\\ p^{*}_{1}\alpha_{1}\\\
p^{*}_{1}\alpha_{1}^{\prime}\\\ \end{bmatrix}$

$\bm{\Sigma}=\begin{bmatrix}\bm{\Sigma}_{11}&\bm{\Sigma}_{12}&\bm{\Sigma}_{13}&\bm{\Sigma}_{14}\\\
\bm{\Sigma}_{21}&\bm{\Sigma}_{22}&\bm{\Sigma}_{23}&\bm{\Sigma}_{24}\\\
\bm{\Sigma}_{31}&\bm{\Sigma}_{32}&\bm{\Sigma}_{33}&\bm{\Sigma}_{34}\\\
\bm{\Sigma}_{41}&\bm{\Sigma}_{42}&\bm{\Sigma}_{43}&\bm{\Sigma}_{44}\\\
\end{bmatrix}\\\
=n\begin{bmatrix}p^{*}_{0}\alpha_{0}-(p^{*}_{0}\alpha_{0})^{2}&{(p^{*}_{0})}^{2}\alpha_{0}\alpha_{0}^{\prime}&p^{*}_{0}p^{*}_{1}\alpha_{0}\alpha_{1}&p^{*}_{0}p^{*}_{1}\alpha_{0}\alpha_{1}^{\prime}\\\
{(p^{*}_{0}})^{2}\alpha_{0}\alpha_{0}^{\prime}&p^{*}_{0}\alpha_{0}^{\prime}-(p^{*}_{0}\alpha_{0}^{\prime})^{2}&p^{*}_{0}p^{*}_{1}\alpha_{0}^{\prime}\alpha_{1}&p^{*}_{0}p^{*}_{1}\alpha_{0}^{\prime}\alpha_{1}^{\prime}\\\
p^{*}_{0}p^{*}_{1}\alpha_{0}\alpha_{1}&p^{*}_{0}p^{*}_{1}\alpha_{0}^{\prime}\alpha_{1}&p^{*}_{1}\alpha_{1}-(p^{*}_{1}\alpha_{1})^{2}&(p^{*}_{1})^{2}\alpha_{1}\alpha_{1}^{\prime}\\\
p^{*}_{0}p^{*}_{1}\alpha_{0}\alpha_{1}^{\prime}&p^{*}_{0}p^{*}_{1}\alpha_{0}^{\prime}\alpha_{1}^{\prime}&(p^{*}_{1})^{2}\alpha_{1}\alpha_{1}^{\prime}&p^{*}_{1}\alpha_{1}^{\prime}-(p^{*}_{1}\alpha_{1}^{\prime})^{2}\\\
\end{bmatrix}$

With this, we note that the marginal distribution of this multivariate
Gaussian distribution gives us a univariate (one-dimensional) Gaussian
distribution for the count of each output in ${\mathbf{c}}^{T}$. For example,
the distribution of the count for event $c_{0|0}$, denoted by $N_{c_{0|0}}$,
can be modeled as $N_{c_{0|0}}\sim\mathcal{N}(\bm{\mu}_{1},\bm{\Sigma}_{11})$.
Then, we find the total rate of data points labeled as class $i$ when labeling
$n$ generated images using the normalized sum of the related random variables
i.e. $\hat{p}_{i}=\frac{1}{n}\sum_{j}N_{{c}_{i|j}}$. More specifically:

$\displaystyle\hat{p}_{0}=\frac{1}{n}(\mathbf{N}_{{\mathbf{c}}_{0|0}}+\mathbf{N}_{{\mathbf{c}}_{0|1}})\sim\bm{\mathcal{N}}(\tilde{\mu}_{\hat{p}_{0}},\tilde{\sigma}^{2}_{\hat{p}_{0}})$
$\displaystyle\tilde{\mu}_{\hat{p}_{0}}=(p^{*}_{0}\alpha_{0}+p^{*}_{1}\alpha_{1}^{\prime})$
$\displaystyle\tilde{\sigma}^{2}_{\hat{p}_{0}}=\frac{1}{n}((p^{*}_{0}\alpha_{0}-(p^{*}_{0}\alpha_{0})^{2})+(p^{*}_{1}\alpha_{1}^{\prime}-(p^{*}_{1}\alpha_{1}^{\prime})^{2})+2{p^{*}_{0}p^{*}_{1}\alpha_{0}\alpha_{1}^{\prime}})$
$\displaystyle{\hat{p}_{1}}=\frac{1}{n}(\mathbf{N}_{{\mathbf{c}}_{1|0}}+\mathbf{N}_{{\mathbf{c}}_{1|1}})\sim\mathcal{N}(\tilde{\mu}_{\hat{p}_{1}},\tilde{\sigma}^{2}_{\hat{p}_{1}})$
$\displaystyle\tilde{\mu}_{\hat{p}_{1}}=(p^{*}_{0}\alpha_{0}^{\prime}+p^{*}_{1}\alpha_{1})$
$\displaystyle\tilde{\sigma}^{2}_{\hat{p}_{1}}=\frac{1}{n}((p^{*}_{0}\alpha_{0}^{\prime}-(p^{*}_{0}\alpha_{0}^{\prime})^{2})+(p^{*}_{1}\alpha_{1}-(p^{*}_{1}\alpha_{1})^{2})+2{p^{*}_{0}p^{*}_{1}\alpha_{0}^{\prime}\alpha_{1}})$

Remark: In Sec 4.1 of the main paper, considering the probability tree diagram
in Fig.1(b) (main paper), we proposed a distribution for the possible events
of classification ($c_{i|j}$), and used it to compute distribution of each
event, and finally the distribution of the output of the sensitive attribute
classifier ($\hat{p}_{0}$, and $\hat{p}_{1}$). Here, we provide more
information on the necessary assumptions and the expanded forms of the
equations. In the following Sec. A.2, we will similarly provide more
information on proposed CLEAM, presented in Sec. 4.2 of the main paper, which
utilizes this statistical model to mitigate the sensitive attribute
classifier’s error.

###  A.2 Additional Details on CLEAM Algorithm

MLE value of Population Mean. In this section, first, we discuss the maximum
likelihood estimate (MLE) of the population mean for a Gaussian distribution.
Given a Gaussian distribution with the population mean
$\tilde{\mu}_{\hat{p}_{0}}$ and standard deviation
$\tilde{\sigma}_{\hat{p}_{0}}$, we can first find the joint probability
distribution from the product of each probabilistic outcome (we introduce the
natural log as a monotonic function, for ease of calculation). Then, to find
the MLE of $\tilde{\mu}_{\hat{p}_{0}}$, we take the partial derivative of this
joint distribution w.r.t. $\tilde{\mu}_{\hat{p}_{0}}$, and solve for its
maximum value. This maximum value is equal to the sample mean,
$\ddot{\mu}_{\hat{p}_{0}}$, as detailed below:

$\displaystyle\frac{\partial}{\partial\tilde{\mu}_{\hat{p}_{0}}}\prod_{i=1}^{s}ln(\frac{1}{\tilde{\sigma}_{\hat{p}_{0}}\sqrt{2\pi}}e^{\frac{-(\hat{p}^{i}_{0}-\tilde{\mu}_{\hat{p}_{0}})^{2}}{2\tilde{\sigma}_{\hat{p}_{0}}^{2}}})=0$
$\displaystyle\frac{1}{\tilde{\sigma}_{\hat{p}_{0}}^{2}}\sum_{i=0}^{s}(\hat{p}^{i}_{0}-\tilde{\mu}_{\hat{p}_{0}})=0$
$\displaystyle\tilde{\mu}_{\hat{p}_{0}}=\frac{1}{s}\sum_{i}^{s}\hat{p}^{i}_{0}=\ddot{\mu}_{\hat{p}_{0}}$

Point Estimate of CLEAM. From this, given that $s$ is sufficiently large, we
utilize the sample mean as the maximum likelihood approximate of the
population mean. As the population mean was modeled in Sec. A.1, we can equate
the sample mean to the expanded theoretical model:

$\displaystyle\ddot{\mu}_{\hat{p}_{0}}=\tilde{\mu}_{\hat{p}_{0}}=p^{*}_{0}\alpha_{0}+(1-p^{*}_{0})\alpha^{\prime}_{1}$

Now given that the classifier’s accuracy
${\bm{\alpha}}=[\alpha_{0},\alpha_{1}]$ and the sample mean
$\ddot{\mu}_{\hat{p}_{0}}$ can be measured, we are able to solve for the
maximum likelihood point estimate of $p^{*}_{0}$, which we denoted with
$\mu_{\texttt{CLEAM}}$ as follows:

$\displaystyle\mu_{\texttt{CLEAM}}=\frac{\ddot{\mu}_{\hat{p}_{0}}-\alpha^{\prime}_{1}}{\alpha_{0}-\alpha^{\prime}_{1}}=\frac{\ddot{\mu}_{\hat{p}_{0}}-1+\alpha_{1}}{\alpha_{0}-1+\alpha_{1}}$

Note that we compute $\mu_{\texttt{CLEAM}}$ w.r.t. $p^{*}_{0}$ i.e.
$\mu_{\texttt{CLEAM}}(p^{*}_{0})$ through-out this paper for ease of
discussion, however as $p^{*}_{1}=1-p^{*}_{0}$, a similar
$\mu_{\texttt{CLEAM}}$ w.r.t. $p^{*}_{1}$ i.e.
$\mu_{\texttt{CLEAM}}(p^{*}_{1})$ can be found with
$\mu_{\texttt{CLEAM}}(p^{*}_{1})=1-\mu_{\texttt{CLEAM}}(p^{*}_{0})$.

Interval Estimate of CLEAM. We acknowledge that there exist other
statistically probable solutions for $\bm{p^{*}}$ that could output the $s$
$\bm{\hat{p}}$ samples, other than the Maximum likelihood point estimate of
$\bm{p^{*}}$. We thus propose the following approximation for the $95\%$
confidence interval of $\bm{p^{*}}$. Recall the notations
$\ddot{\mu}_{\hat{p}_{0}}$ and $\ddot{\sigma}_{\hat{p}_{0}}$ are the sample
mean and standard deviation respectively:

$\ddot{\mu}_{\hat{p}_{0}}=\frac{1}{s}\sum_{i}^{s}\hat{p}^{i}_{0}\quad;\quad\ddot{\sigma}_{\hat{p}_{0}}=\sqrt{\frac{\sum^{s}_{i=1}(\hat{p}^{i}_{0}-\ddot{\mu}_{\hat{p}_{0}})^{2}}{s-1}}$

Since $\bm{\hat{p}}$ follows a Gaussian distribution, we can propose the
following equation:

$\displaystyle
Pr(-z_{\frac{\delta}{2}}\leq\frac{\ddot{\mu}_{\hat{p}_{0}}-\tilde{\mu}_{\hat{p}_{0}}}{\frac{\ddot{{\sigma}}_{\hat{p}_{0}}}{\sqrt{s}}}\leq{\mathbf{z}}_{\frac{\delta}{2}})=1-\delta$

Solving for $\tilde{\mu}_{\hat{p}}$, we get:

$\displaystyle
Pr(\ddot{\mu}_{\hat{p}_{0}}+z_{\frac{\delta}{2}}(\frac{\ddot{\sigma}_{\hat{p}_{0}}}{\sqrt{s}})\geq\tilde{\mu}_{\hat{p}_{0}}\geq\ddot{\mu}_{\hat{p}_{0}}-z_{\frac{\alpha}{2}}\frac{\ddot{\sigma}_{\hat{p}_{0}}}{\sqrt{s}})=1-\delta$

Then, given that
$\tilde{\mu}_{\hat{p}}=p^{*}_{0}\alpha_{0}+p^{*}_{1}\alpha_{1}^{\prime}=p^{*}_{0}(\alpha_{0}-\alpha_{1}^{\prime})+\alpha_{1}^{\prime}$
we formulate the following:

$\displaystyle
Pr(\frac{\ddot{\mu}_{\hat{p}_{0}}+z_{\frac{\delta}{2}}(\frac{\ddot{\sigma}_{\hat{p}_{0}}}{\sqrt{s}})-\alpha_{1}^{\prime}}{\alpha_{0}-\alpha_{1}^{\prime}}\geq
p^{*}_{0}\geq\frac{\ddot{\mu}_{\hat{p}_{0}}-z_{\frac{\alpha}{2}}\frac{\ddot{\sigma}_{\hat{p}_{0}}}{\sqrt{s}}-\alpha_{1}^{\prime}}{\alpha_{0}-\alpha_{1}^{\prime}})=1-\delta$
(11)

As such when $\delta=0.05$, we can determine that the $95\%$ approximated
confidence interval of $p^{*}_{0}$ is :

$\displaystyle\rho_{\texttt{CLEAM}}(p^{*}_{0})=[\mathcal{L}(p^{*}_{0}),\mathcal{U}(p^{*}_{0})]=[\frac{\ddot{\mu}_{\hat{p}_{0}}-1.96(\frac{\ddot{\sigma}_{\hat{p}_{0}}}{\sqrt{s}})-\alpha_{1}^{\prime}}{\alpha_{0}-\alpha_{1}^{\prime}}\quad,\quad\frac{\ddot{\mu}_{\hat{p}_{0}}+1.96\frac{\ddot{\sigma}_{\hat{p}_{0}}}{\sqrt{s}}-\alpha_{1}^{\prime}}{\alpha_{0}-\alpha_{1}^{\prime}}]$

Extending the point estimate to a multiple label setup. We remark that in
current literature, fairness of generative models has been studied for binary
sensitive attributes mainly due to the lack of availability of a large labeled
dataset needed for systematic experimentation. As a result, CLEAM similarly
focuses on binary SA to address a common flaw in the evaluation process of the
many proposed State-of-the-Art methods.

Assuming that the constraint of the dataset is addressed, our same CLEAM
approach can be easily extended to a multi-label setup. For example, given a 3
label sensitive attribute where $p^{*}_{j}$ is the probability of generating a
sample with label $j$ and $\alpha_{i|j}$ denotes the probability (“accuracy”)
of the SA classifier in classifying a sample with GT label $j$ as $i$ for
$i$,$j\in\\{0,1,2\\}$. Fig. 5 shows our statistical model for this setting. We
can then similarly solve for the point estimate by solving the matrix:

$\begin{bmatrix}\alpha_{0|0}&\alpha_{0|1}&\alpha_{0|2}\\\
\alpha_{1|0}&\alpha_{1|1}&\alpha_{1|2}\\\
\alpha_{2|0}&\alpha_{2|1}&\alpha_{2|2}\\\
\end{bmatrix}\begin{bmatrix}p^{*}_{0}\\\ p^{*}_{1}\\\
p^{*}_{2}\end{bmatrix}=\begin{bmatrix}\ddot{\mu}_{\hat{p}_{0}}\\\
\ddot{\mu}_{\hat{p}_{1}}\\\ \ddot{\mu}_{\hat{p}_{2}}\end{bmatrix}$ (12)

Figure 5:  Our statistical model for fairness measurement when considering a
multi-label SA. For illustration purposes, we utilize 3 labels for a given SA.
Note that, our same approach can be applied to other multi-label settings.
This statistical model accounts for inaccuracies in the SA classifier and is
the base of our proposed CLEAM (see Sec. 4.1). Here, $p^{*}_{j}$ is the ground
truth probability of a generator outputting a sample with label $j$ and
${\alpha}_{i|j}$ denotes the probability (“accuracy”) of the SA classifier
classifying a sample with GT label $j$ as $i$ for $i,j\in\\{0,1,2\\}$.

###  A.3 Details on Fairness Metric

Fairness in generative models is defined as Equal Representation meaning that
the generator is supposed to generate an equal number of samples for each
element of an attribute, e.g., an equal number of generated Male and Female
samples when the sensitive attribute is Gender. Therefore, the expected
distribution for a fair generator is usually a uniform distribution denoted by
$\bm{\bar{p}}$. Considering this, the fairness discrepancy (FD) metric [1]
measures the L2 norm between $\bm{\bar{p}}$ and the estimated class
probability of the generator by each measurement method, i.e.
$\bm{\mu}_{\dagger}$, where $\dagger\in${Base, CLEAM, Div}, as follows:

$f=|\bm{\bar{p}}-\bm{\mu}_{\dagger}|_{2}$ (13)

Note that for a fair generator the fairness discrepancy $f$ would be zero,
which also indicates zero bias.

###  A.4 Details of Significance of the Baseline Errors

In the main manuscript (Sec. 3 of the main paper), we discussed that the
relative improvement of the previous fair generative models could be small,
e.g. Teo et al. [2] and Um et al. [12] report a relative improvement in the
fairness of 0.32% and 0.75%, compared to imp-weighting [1], and they fall
within the range of our experiment’s smallest relative error,
$e_{\mu_{\texttt{Base}}}$=4.98%. Here, we provide more detail on how we
calculate the relative improvements in the main manuscript. Specifically, we
calculate the relative change of the proposed work against the previous work
with the following:

$\displaystyle
Relaitve\;Improvement=\frac{|(\hat{p}_{0}\;of\;previous\;work)-(\hat{p}_{0}\;of\;proposed\;work)|}{(\hat{p}_{0}\;of\;previous\;work)}$
(14)

Notice that this is similar to $e_{\mu_{\texttt{Base}}}$ of Eqn. 1 in the main
paper. For example, Teo et al. (Tab. 1 90_10 and perc=0.1 settings) [2]
reports that fairTL measures a $f=0.105$ which is compared against the
previous work’s (Choi et al. [1]) $f=0.107$. Utilizing Eqn. 13, we find that
this is equivalent to $p^{*}_{0}=$ 0.4257 or 0.5743, and 0.4243 or 0.5757,
respectively. We remark that here we report two values per $f$, as the FD
metric is a symmetric metric. Then applying Eqn. 14, and taking the maximum of
the values, we find the relative improvement to be $0.32\%$, at best. Note
that as we mentioned in the main paper for this setup the baseline measurement
framework results in $4.98\%$ error rate (with the best performing sensitive
attribute classifier), meaning that it may not be reliable for gauging the
improvement.

##  Appendix B Deeper Analysis on Error in Fairness Measurement

In the main paper Sec.3, we discussed that there could be considerable error
in the fairness measurement, $\bm{\hat{p}}$, even though the sensitive
attribute classifier’s accuracy is considerably high. In addition, we further
develop on this and discuss two additional factors that could result in a
variation of errors. We remark that in the main manuscript, we report
diversity only using VGG-16, as specified by Keswani et al. [46]. Further
discussion in Sec. D.2

Accuracy of Individual Classes (${\bm{\alpha}}=\\{\alpha_{0},\alpha_{1}\\}$)
Impacts the Degree of Error. Notice that in some cases even though the
sensitive attribute classifier may have a very similar average accuracy,
different degrees of errors could exist for the two different classifiers e.g.
R18 and R34 in Tab. 3. This is because the fairness measurement error is not
only dependent on the average accuracy but on the individual class accuracy
i.e. $\alpha_{0}$ and $\alpha_{1}$. More specifically, given that there is a
larger error in $\alpha_{0}$ for R34 and the bias exists in $p^{*}_{0}=0.643$,
this results in a compounded effect and hence a larger error of
$e_{\mu_{\texttt{Base}}}$=11.98% is observed as compared to R18
$e_{\mu_{\texttt{Base}}}$=6.84%.

Uniform Inaccuracies at Unbiased Test-Point ($p^{*}_{0}=p^{*}_{1}=$0.5). In
our extended experiments in Sec. D.3 for a Pseudo-generator, we discuss that
for some sensitive attribute classifiers e.g. ResNet-18 for Gender and
BlackHair, the Baseline performs better than CLEAM at the unbiased test-point
i.e. $p^{*}_{0}=0.5$. This is just due to the Gender and Blackhair setups
having a specific combination of (i) the Pseudo-Generator producing almost
perfectly unbiased data with $\bm{p^{*}}=[0.5,0.5]$, (ii) sensitive attribute
classifier with almost perfectly uniform inaccuracies
$\alpha_{0}^{\prime}\approx\alpha_{1}^{\prime}$, thereby leading to uniform
misclassification and hence the false impression of better accuracy by the
baseline method, at $\bm{p^{*}}=[0.5,0.5]$ (See Tab. 4 for extracted table) .
To further illustrate this, notice how the ResNet-18 trained on Cat/Dog did
not demonstrate this better performance in the Baseline due to its non-uniform
${\bm{\alpha}}$. Nevertheless, we note this situation whereby the Baseline
outperform CLEAM is specific to the test-point $p^{*}_{0}=0.5$ and does not
impact the overall effectiveness of CLEAM. Furthermore, CLEAM still
demonstrates outstanding results with low error for both the PE and IE at
$p^{*}_{0}=0.5$.

To further demonstrate these effects, we repeat this same experiment, but with
sensitive attributes Young and Attractive from the CelebA dataset. As seen in
Tab. 5, both Young or Attractive have similar average accuracy,
$\alpha_{Avg}=\frac{\alpha_{0}+\alpha_{1}}{2}$ of $0.801$ and $0.794$ but a
different $skew_{\bm{\alpha}}=|\alpha_{0}-\alpha_{1}|$ of 0.103 and 0.027. As
such, we are able to investigate the effects that $skew_{\bm{\alpha}}$ has on
both CLEAM and Baseline. We did not include Diversity in this study, due to
its poor performance on harder sensitive attribute, as discussed in Sec. D.2.
From our results in Tab. 5, we observe that as the $skew_{\bm{\alpha}}$
increases from sensitive attribute Attractive to Young, the error becomes much
more significant in the baseline method. The average $e_{\mu_{\texttt{Base}}}$
increases from $12.69\%$ to $17.63\%$. Furthermore, unlike Gender and
Blackhair, who have relatively negligible skew, Young and Attractive observes
a significantly larger error at $\bm{p^{*}}=[0.5,0.5]$.

Table 3:  Extracted from Tab. 13 for ease of viewing. Comparing the point
estimates and interval estimates of Baseline [1] and our proposed CLEAM
measurement framework in estimating $\bm{p^{*}}$ of the GenData datasets
sampled from (A) StyleGAN2 [3]. The $p^{*}_{0}$ value for each GAN with a
certain attribute is determined by manually hand-labeling the generated data.
We utilize two different classifiers Resnet-18/34 (R18, R34)[34] with
different accuracy ${\bm{\alpha}}$ to obtain $\bm{\hat{p}}$ by classifying
samples w.r.t. BlackHair. For calculating each $\bm{\hat{p}}$, we utilize
$n=400$ samples and evaluate for a batch size of $s=30$. We repeat this for 5
experimental runs and report the mean error rate, per Eqn. 1 of the main
paper.

|  |  | Point Estimate | Interval Estimate
---|---|---|---|---
Classifier | ${\bm{\alpha}}=\\{\alpha_{0},\alpha_{1}\\}$ | Avg. ${\bm{\alpha}}$ | Baseline | Diversity | CLEAM (Ours) | Baseline | Diversity | CLEAM (Ours)
|  |  | $\mu_{\texttt{Base}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{Div}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{CLEAM}}$ |  $e_{\mu}(\downarrow)$ | $\rho_{\texttt{Base}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{Div}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{CLEAM}}$ | $e_{\rho}(\downarrow)$
(A) StyleGAN2
BlackHair with GT class probability $p^{*}_{0}$=0.643 |
R18 | {0.869, 0.885} | 0.88 | 0.599 | 6.84% | — | — | 0.641 |  0.31% | [0.591, 0.607] | 8.08% | — | — | [0.631, 0.652] | 1.40%
R34 | {0.834, 0.916} | 0.88 | 0.566 | 11.98% | — | — | 0.644 |  0.16% | [0.561, 0.572] | 12.75% | — | — | [0.637, 0.651] | 1.24%

Table 4:  Extracted from Tab. 13 for ease of viewing. Comparing the point
estimates and interval estimate of Baseline [1], Diversity [46] and proposed
CLEAM measurement frameworks in estimating different $\bm{p^{*}}$ of a pseudo-
generator, based on the CelebA [47] and AFHQ [48] dataset. The $\bm{\hat{p}}$
is computed with a ResNet-18 sensitive attribute classifier and the error rate
is reported using Eqn. 1 of the main paper. We repeat this for Gender,
BlackHair and Cat/Dog attributes.

| Point Estimate | Interval Estimate
---|---|---
GT | Baseline | Diversity | CLEAM (Ours) | Baseline | Diversity | CLEAM (Ours)
| $\mu_{\texttt{Base}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{Div}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{CLEAM}}$ |  $e_{\mu}(\downarrow)$ | $\rho_{\texttt{Base}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{Div}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{CLEAM}}$ | $e_{\rho}(\downarrow)$
${\bm{\alpha}}$=[0.976,0.979], Gender (CelebA)
$p^{*}_{0}$=0.5 | 0.501 |  0.20% | 0.481 | 3.80% | 0.502 | 0.40% | [0.495 , 0.507 ] |  1.40% | [0.473 , 0.490 ] | 5.40% | [0.497, 0.508] | 1.60%
${\bm{\alpha}}$=[0.881,0.887], BlackHair (CelebA)
$p^{*}_{0}$=0.5 | 0.500 |  0.00% | 0.521 | 4.20% | 0.504 | 0.8% | [ 0.495 , 0.505 ] |  1.00% | [0.506 , 0.536 ] | 7.20% | [0.497, 0.511] | 2.20%
${\bm{\alpha}}$=[0.953,0.0.990], Cat/Dog (AFHQ)
$p^{*}_{0}$=0.5 | 0.486 | 2.80% | 0.469 | 6.20% | 0.505 |  1.00% | [ 0.480 , 0.493 ] | 4.00% | [ 0.458, 0.480 ] | 8.40% | [ 0.498 , 0.511 ] |  2.20%

Table 5: Duplicate of Tab. 14 for ease of viewing. Comparing point estimate
and interval estimate of Baseline [1], and proposed CLEAM measurement
framework on a pseudo-generator with sensitive attribute
$\\{\texttt{Young},\texttt{Attractive}\\}$

| Point Estimate | Interval Estimate
---|---|---
GT | Baseline | Diversity | CLEAM (Ours) | Baseline | Diversity | CLEAM (Ours)
| $\mu_{\texttt{Base}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{Div}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{CLEAM}}$ |  $e_{\mu}(\downarrow)$ | $\rho_{\texttt{Base}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{Div}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{CLEAM}}$ | $e_{\rho}(\downarrow)$
${\bm{\alpha}}$=[0.749,0.852], Young
$p^{*}_{0}=0.9$ | 0.690 | 23.33% | — | — | 0.905 |  0.56% | [0.684,0.695] | 24.00% | — | — | [0.890,0.920] |  2.22%
$p^{*}_{0}=0.8$ | 0.630 | 21.25% | — | — | 0.804 |  0.50% | [0.625,0.635] | 21.88% | — | — | [0.795,0.813] |  1.63%
$p^{*}_{0}=0.7$ | 0.570 | 18.57% | — | — | 0.698 |  0.29% | [0.565,0.575] | 19.29% | — | — | [0.690,0.706] |  1.43%
$p^{*}_{0}=0.6$ | 0.510 | 15.00% | — | — | 0.595 |  0.83% | [0.505,0.515] | 15.83% | — | — | [0.590,0.600] |  1.67%
$p^{*}_{0}=0.5$ | 0.450 | 10.0% | — | — | 0.506 |  1.20% | [0.445,0.455] | 11.00% | — | — | [0.502,0.510] |  2.00%
Avg Error | 17.63% |  | —% |  |  0.68% |  | 18.40% |  | —% |  |  1.79%
${\bm{\alpha}}$=[0.780,0.807], Attractive
$p^{*}_{0}=0.9$ | 0.730 | 18.89% | — | — | 0.908 |  0.89% | [0.724,0.736] | 19.56% | — | — | [0.900,0.916] |  1.78%
$p^{*}_{0}=0.8$ | 0.670 | 16.25% | — | — | 0.804 |  0.50% | [0.665,0.675] | 16.88% | — | — | [0.795,0.813] |  1.63%
$p^{*}_{0}=0.7$ | 0.600 | 14.29% | — | — | 0.696 |  0.57% | [0.594,0.606] | 15.14% | — | — | [0.690,0.712] |  1.71%
$p^{*}_{0}=0.6$ | 0.540 | 10.00% | — | — | 0.592 |  1.33% | [0.534,0.546] | 11.00% | — | — | [0.580,0.604] |  3.33%
$p^{*}_{0}=0.5$ | 0.480 | 4.00% | — | — | 0.493 |  1.40% | [0.475,0.485] | 5.00% | — | — | [0.487,0.499] |  2.60%
Avg Error | 12.69% |  | —% |  |  0.94% |  | 13.52% |  | —% |  |  2.22%

##  Appendix C Validating Statistical Model for Classifier Output

###  C.1  Validation of Sample-Based Estimate vs Model-Based Estimate

As described in the main paper, we utilize the sample-based estimate,
$\ddot{\mu}_{\hat{p}_{0}}$, ${\ddot{\sigma}^{2}_{\hat{p}_{0}}}$ as an
approximate for the model-based estimate $\tilde{\mu}_{\hat{p}_{0}}$,
${\tilde{\sigma}^{2}_{\hat{p}_{0}}}$. As discussed in Sec. A.2,
$\ddot{\mu}_{\hat{p}_{0}}$ allows us to find the maximum likelihood
approximate of $\bm{p^{*}}$.

To validate this approximation,  we utilize a ResNet-18 trained on Gender and
BlackHair to compute $\bm{\hat{p}}$. Then with the samples from the pseudo-
generators with different $\bm{p^{*}}$ (following Sec. D.3) we computed
$\bm{\hat{p}}$ with a batch-size of $s=30$ and sample size $n=400$. Finally,
we calculate the sample-based estimates as given in Eqn. 6, 7 of the main
paper. As the GT $\bm{p^{*}}$ and classifier’s accuracy ${\bm{\alpha}}$ is
known, we also calculate the model-based estimates as given in Eqn. 4, 5 of
the main manuscript and compare it against the sample-based estimates.

Our results in Tab. 6 shows that both the sample and theoretical means and
standard deviations are close approximate to one another. Thus, we can utilise
the sample statistics as a close approximation in our proposed method, CLEAM.
Additional results for different values of batch-sizes ($s$) and sample-sizes
($n$) are tabulated in Tab. 7, 8 and 9. Notice that a reduction in $s$ and $n$
values contributed to increased errors between the sample-based and model-
based estimates. While making $s$ very large ($s=200$), results in the sample
based estimate almost a perfectly approximating the model based estimates.

Table 6:  Comparing sample-based estimates
($\ddot{\bm{\mu}}_{\bm{\hat{p}}_{0}}$,
$\ddot{\bm{\sigma}}_{\bm{\hat{p}}_{0}}$) against model-based estimates
($\tilde{\bm{\mu}}_{\bm{\hat{p}}_{0}}$,
$\tilde{\bm{\sigma}}_{\bm{\hat{p}}_{0}}$). The results show that sample-based
estimates are close to model-based estimates. Furthermore, note the
discrepancy between $p^{*}_{0}$ and $\ddot{\mu}_{\hat{p}_{0}}$, and that
between $p^{*}_{0}$ and $\tilde{\mu}_{\hat{p}_{0}}$, highlighting the issue of
using $\hat{p}_{0}$ directly to estimate $p^{*}_{0}$ and the need to
compensate for the sensitive attribute classifier error as we discussed. We
utilize a $s=30$ and $n=400$.

GT | Sampled-based estimates | Model-based estimates
---|---|---
| $\ddot{\mu}_{\hat{p}_{0}}$ |  $\sqrt{\ddot{\sigma}^{2}_{\hat{p}_{0}}}$ | $\tilde{\mu}_{\hat{p}_{0}}$ | $\sqrt{\tilde{\sigma}^{2}_{\hat{p}_{0}}}$
Gender, ${\bm{\alpha}}$=[0.976,0.979]
$p^{*}_{0}=0.9$ | 0.881 | 0.0101 | 0.881 | 0.0106
$p^{*}_{0}=0.8$ | 0.781 | 0.0133 | 0.785 | 0.0135
$p^{*}_{0}=0.7$ | 0.692 | 0.0149 | 0.690 | 0.0152
$p^{*}_{0}=0.6$ | 0.590 | 0.0165 | 0.594 | 0.0162
$p^{*}_{0}=0.5$ | 0.503 | 0.0164 | 0.499 | 0.0164
${\bm{\alpha}}$=[0.881,0.887], Black-Hair
$p^{*}_{0}=0.9$ | 0.802 | 0.0130 | 0.804 | 0.0139
$p^{*}_{0}=0.8$ | 0.723 | 0.0151 | 0.727 | 0.0162
$p^{*}_{0}=0.7$ | 0.653 | 0.0169 | 0.650 | 0.0177
$p^{*}_{0}=0.6$ | 0.580 | 0.0180 | 0.574 | 0.0186
$p^{*}_{0}=0.5$ | 0.502 | 0.0180 | 0.497 | 0.0189

Table 7: We repeat the same experiment as Tab.6 with $s=20$ and $n=400$
samples.

GT | Sampled-based estimates | Model-based estimates
---|---|---
| $\ddot{\mu}_{\hat{p}_{0}}$ |  $\sqrt{\ddot{\sigma}^{2}_{\hat{p}_{0}}}$ | $\tilde{\mu}_{\hat{p}_{0}}$ | $\sqrt{\tilde{\sigma}^{2}_{\hat{p}_{0}}}$
Gender, ${\bm{\alpha}}$=[0.976,0.979]
$p^{*}_{0}=0.9$ | 0.855 | 0.0201 | 0.881 | 0.0106
$p^{*}_{0}=0.8$ | 0.774 | 0.0211 | 0.785 | 0.0135
$p^{*}_{0}=0.7$ | 0.672 | 0.0219 | 0.690 | 0.0152
$p^{*}_{0}=0.6$ | 0.580 | 0.0181 | 0.594 | 0.0162
$p^{*}_{0}=0.5$ | 0.510 | 0.0230 | 0.499 | 0.0164
${\bm{\alpha}}$=[0.881,0.887], Black-Hair
$p^{*}_{0}=0.9$ | 0.768 | 0.180 | 0.804 | 0.0139
$p^{*}_{0}=0.8$ | 0.712 | 0.210 | 0.727 | 0.0162
$p^{*}_{0}=0.7$ | 0.658 | 0.190 | 0.650 | 0.0177
$p^{*}_{0}=0.6$ | 0.554 | 0.230 | 0.574 | 0.0186
$p^{*}_{0}=0.5$ | 0.508 | 0.242 | 0.497 | 0.0189

Table 8: We repeat the same experiment as per Tab.6 with $s=30$ and $n=200$
samples.

GT | Sampled-based estimates | Model-based estimates
---|---|---
| $\ddot{\mu}_{\hat{p}_{0}}$ |  $\sqrt{\ddot{\sigma}^{2}_{\hat{p}_{0}}}$ | $\tilde{\mu}_{\hat{p}_{0}}$ | $\sqrt{\tilde{\sigma}^{2}_{\hat{p}_{0}}}$
Gender, ${\bm{\alpha}}$=[0.976,0.979]
$p^{*}_{0}=0.9$ | 0.860 | 0.0232 | 0.881 | 0.0149
$p^{*}_{0}=0.8$ | 0.780 | 0.0286 | 0.785 | 0.0191
$p^{*}_{0}=0.7$ | 0.710 | 0.0294 | 0.690 | 0.0215
$p^{*}_{0}=0.6$ | 0.578 | 0.0380 | 0.594 | 0.0228
$p^{*}_{0}=0.5$ | 0.520 | 0.0321 | 0.499 | 0.0233
${\bm{\alpha}}$=[0.881,0.887], Black-Hair
$p^{*}_{0}=0.9$ | 0.742 | 0.0312 | 0.804 | 0.0197
$p^{*}_{0}=0.8$ | 0.740 | 0.0332 | 0.727 | 0.0229
$p^{*}_{0}=0.7$ | 0.610 | 0.0291 | 0.650 | 0.0250
$p^{*}_{0}=0.6$ | 0.582 | 0.350 | 0.574 | 0.0262
$p^{*}_{0}=0.5$ | 0.542 | 0.388 | 0.497 | 0.0267

Table 9: We repeat the same experiment as per Tab.6 with $s=200$ and $n=400$
samples.

GT | Sampled-based estimates | Model-based estimates
---|---|---
| $\ddot{\mu}_{\hat{p}_{0}}$ |  $\sqrt{\ddot{\sigma}^{2}_{\hat{p}_{0}}}$ | $\tilde{\mu}_{\hat{p}_{0}}$ | $\sqrt{\tilde{\sigma}^{2}_{\hat{p}_{0}}}$
Gender, ${\bm{\alpha}}$=[0.976,0.979]
$p^{*}_{0}=0.9$ | 0.881 | 0.0104 | 0.881 | 0.0106
$p^{*}_{0}=0.8$ | 0.784 | 0.0133 | 0.785 | 0.0135
$p^{*}_{0}=0.7$ | 0.690 | 0.0153 | 0.690 | 0.0152
$p^{*}_{0}=0.6$ | 0.594 | 0.0160 | 0.594 | 0.0162
$p^{*}_{0}=0.5$ | 0.500 | 0.0164 | 0.499 | 0.0164
${\bm{\alpha}}$=[0.881,0.887], Black-Hair
$p^{*}_{0}=0.9$ | 0.804 | 0.0137 | 0.804 | 0.0139
$p^{*}_{0}=0.8$ | 0.726 | 0.0160 | 0.727 | 0.0162
$p^{*}_{0}=0.7$ | 0.650 | 0.0179 | 0.650 | 0.0177
$p^{*}_{0}=0.6$ | 0.573 | 0.0185 | 0.574 | 0.0186
$p^{*}_{0}=0.5$ | 0.498 | 0.0191 | 0.497 | 0.0189

###  C.2 Goodness-of-Fit Test: $\bm{\hat{p}}$ from the Real GANs with Our
Theoretical Model

In order to make sure that our proposed theoretical model in Eqn. 4 and Eqn. 5
of the main paper, is also a good representation of the $\bm{\hat{p}}$
distribution when using a generator, we perform a goodness of fit test between
the proposed model for the distribution of $\bm{\hat{p}}$ and sample data
generated by a GAN.

Table 10:  Validating goodness-of-fit of the proposed theoretical model
against generated samples. A KS-test [57] is conducted between the samples
distribution of $\bm{\hat{p}}$ \- measured from GenData with a ResNet-18, and
the theoretical distribution of $\bm{\hat{p}}$. We utilize $s$=30, $n$=400
with $D_{crit}$=0.24. Since $\eta<D_{crit}$, all of the $\bm{\hat{p}}$ are
statistically similar to the theoretical Gaussian at 95% confidence. This is
further observed by the sample-based mean ($\ddot{\mu}$) $\approx$ model-based
mean ($\tilde{\mu}$).

Model Type | Sensitive Attribute |  $\eta$ | $\tilde{\mu}$ | $\ddot{\mu}$
---|---|---|---|---
StyleGAN2 | Gender | 0.1048 | 0.610 | 0.609
Blackhair | 0.1065 | 0.601 | 0.601
StyleSwin | Gender | 0.1509 | 0.628 | 0.629
Blackhair | 0.1079 | 0.619 | 0.614

To do this, we first obtain $s=30$ values of $\bm{\hat{p}}$ from framework
shown in Fig. 1 of the main paper, and use StyleGAN2 [3] and StyleSwin [4] as
the generative model. Then using ResNet-18 with known ${\bm{\alpha}}$ and
GAN’s GT $\bm{p^{*}}$, as discussed in Sec. 4.1 of the main paper, we form the
theoretical model’s Gaussian distribution,
$\mathcal{N}(\tilde{\mu}_{\hat{p}_{0}},\tilde{\sigma}^{2}_{\hat{p}_{0}})$.

Now with both our model distribution and the GAN samples, we utilise the
Kolmogorov-Smirnov goodness of fit test (K-S test) to determine if the samples
distribution is statistically similar to the proposed Gaussian model. We thus
propose the following hypothesis test for the samples
$\hat{p}^{i}_{j},i\in\\{1,\cdots,s\\}$:

$\displaystyle\mathbf{H_{0}}:\textit{the samples $\hat{p}^{i}_{j}$ belong to
the modelled distribution.}$ $\displaystyle\mathbf{H_{1}}:\textit{at least one
of the samples $\hat{p}^{i}_{j}$ does not match the modelled distribution.}$

The K-S test then measures a D-statistic ($\eta$) and compares it against a
$D_{crit}$ for a given $s$. As we use $s=30$, and a significance level
$\delta=0.05$ in our setup, we have $D_{crit}=0.24$. As seen from Tab. 10, all
of the measured $\eta$ values are below $D_{crit}$, thus we cannot reject the
null hypothesis at a $95\%$ confidence with the K-S test. Therefore, we
conclude that the distribution of the obtained samples from the framework (by
GANs as generator) are statistically similar to the proposed Gaussian
distribution. As a result, we can utilise CLEAM to approximate the
$\bm{p^{*}}$ range in the presence of a real GAN as the generator.

We further perform a Quantile-Quantile(QQ) analysis to provide a more visual
representation. In particular, we plot the Quantile-Quantile(QQ) plot between
the $\bm{\hat{p}}$ samples (produced for the data generated by the GAN) and
proposed model. As seen in Fig. 6, the $\bm{\hat{p}}$ samples from GAN
correlate tightly with the standardised line (in red), a line indicating a
perfect correlation between theoretical and sample quantiles. This analysis
supports our claim that the $\bm{\hat{p}}$ samples from a real generator (GAN)
follow the distribution estimated by the proposed model.

(a) StyleGAN2, Gender.

(b) StyleSwin2, Gender

(c) StyleGAN2, Blackhair

(d) StyleSwin, BlackHair

Figure 6: Quartile-Quartile(QQ) plot between $s=30$ $\bm{\hat{p}}$ samples
calculated for StyleGAN2 [3] and StyleSwin [4] generators and proposed
theoretical model for $\bm{\hat{p}}$

##  Appendix D Additional Experiments

###  D.1 Experimental Results with Standard Deviation

In the main manuscript, we did not include the error bars of our experiments
due to space constraints. Hence, in this section, we provide the full tables
for Tab. 1 and 2 of the main manuscript with the standard deviation over 5
runs. Note that generally, the standard deviation at each test point is
relatively small and hence can be considered as negligible. This is likely due
to the large $s$ and $n$ utilized. As a result, we can utilize the mean
results (as seen in the main manuscript) to compare CLEAM against Diversity
and the Baseline.

Table 11:  Comparing the point estimates and interval estimates of Baseline
[1], Diversity [46] and our proposed CLEAM measurement framework in estimating
$\bm{p^{*}}$ of the GenData datasets sampled from (A) StyleGAN2 [3] and (B)
StyleSwin [4]. The $p^{*}_{0}$ value for each GAN with a certain sensitive
attribute is determined by manually hand-labeling the generated data. We
utilize four different sensitive attribute classifier Resnet-18/34 (R18,
R34)[34], MobileNetv2 (MN2)[35] and VGG-16 (V16)[36], with different accuracy
${\bm{\alpha}}$, to classify attributes Gender and BlackHair, to obtain
$\bm{\hat{p}}$. Each $\bm{\hat{p}}$ utilizes $n=400$ samples and is evaluated
for a batch size of $s=30$. We repeat this for 5 experimental runs and report
the mean error rate, per Eqn. 1 of the main manuscript.

| Point Estimate | Interval Estimate
---|---|---
| Baseline | Diversity | CLEAM (Ours) | Baseline | Diversity | CLEAM (Ours)
| $\mu_{\texttt{Base}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{Div}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{CLEAM}}$ | $e_{\mu}(\downarrow)$ | $\rho_{\texttt{Base}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{Div}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{CLEAM}}$ | $e_{\rho}(\downarrow)$
(A) StyleGAN2
Gender with GT class probability $p^{*}_{0}$=0.642
R18 | 0.610 $\pm$ 0.004 | 4.98% | — | — | 0.638 $\pm$ 0.006 | 0.62% | [0.602$\pm$ 0.004, 0.618$\pm$ 0.004] | 6.23% | — | — | [0.629 $\pm$ 0.006, 0.646$\pm$ 0.006] | 2.02%
R34 | 0.596$\pm$ 0.003 | 7.17% | — | — | 0.634$\pm$ 0.002 | 1.25% | [0.589$\pm$ 0.003, 0.599$\pm$ 0.003] | 8.26% | — | — | [0.628$\pm$ 0.002, 0.638$\pm$ 0.002] | 2.18%
MN2 | 0.607 $\pm$ 0.003 | 5.45% | — | — | 0.637 $\pm$ 0.002 |  0.78% | [0.602 $\pm$ 0.003, 0.612 $\pm$ 0.003] | 6.23% | — | — | [0.632 $\pm$ 0.002, 0.643 $\pm$ 0.002] | 1.56%
V16 | 0.532 $\pm$ 0.007 | 17.13% | 0.550 $\pm$ 0.011 | 14.3% | 0.636 $\pm$ 0.007 |  0.93% | [0.526 $\pm$ 0.007, 0.538 $\pm$ 0.007] | 18.06% | [0.536 $\pm$ 0.011 , 0.564 $\pm$ 0.011] | 16.51% | [0.628 $\pm$ 0.007, 0.644 $\pm$ 0.007] | 2.18%
Avg Error | 8.68% |  | 14.30% |  |  0.90% |  | 9.70% |  | 16.51% |  |  1.99%
BlackHair with GT class probability $p^{*}_{0}$=0.643
R18 | 0.599 $\pm$ 0.006 | 6.84% | — | — | 0.641 $\pm$ 0.004 | 0.31% | [0.591 $\pm$ 0.006, 0.607 $\pm$ 0.005] | 8.08% | — | — | [0.631 $\pm$ 0.004, 0.652 $\pm$ 0.003] | 1.40%
R34 | 0.566 $\pm$ 0.007 | 11.98% | — | — | 0.644 $\pm$ 0.008 |  0.16% | [0.561 $\pm$ 0.007, 0.572 $\pm$ 0.006] | 12.75% | — | — | [0.637 $\pm$ 0.009, 0.651 $\pm$ 0.008] | 1.24%
MN2 | 0.579 $\pm$ 0.007 | 9.95% | — | — | 0.639 $\pm$ 0.007 |  0.62% | [0.574 $\pm$ 0.008, 0.584 $\pm$ 0.008] | 10.73% | — | — | [0.632 $\pm$ 0.007, 0.647 $\pm$ 0.007] | 1.71%
V16 | 0.603 $\pm$ 0.004 | 6.22% | 0.582 $\pm$ 0.011 | 9.49% | 0.640 $\pm$ 0.005 |  0.47% | [0.597 $\pm$ 0.004, 0.608 $\pm$ 0.003] | 7.15% | [0.568 $\pm$ 0.010, 0.596 $\pm$ 0.011] | 11.66% | [0.632 $\pm$ 0.004, 0.648 $\pm$ 0.005] | 1.71%
Avg Error | 8.75% |  | 9.49% |  |  0.39% |  | 9.68% |  | 11.66% |  |  1.52%
(B) StyleSwin
Gender with GT class probability $p^{*}_{0}$=0.656
R18 | 0.620 $\pm$ 0.005 | 5.49% | — | — | 0.648 $\pm$ 0.004 | 1.22% | [0.612 $\pm$ 0.004,0.629 $\pm$ 0.005] | 6.70% | — | — | [0.639 $\pm$ 0.005,0.658 $\pm$ 0.005] | 2.59%
R34 | 0.610 $\pm$ 0.002 | 7.01% | — | — | 0.649 $\pm$ 0.005 |  1.07% | [0.605 $\pm$ 0.003,0.615 $\pm$ 0.003] | 7.77% | — | — | [0.643 $\pm$ 0.006,0.654 $\pm$ 0.006] |  1.98%
MN2 | 0.623 $\pm$ 0.008 | 5.03% | — | — | 0.655 $\pm$ 0.005 |  0.15% | [0.618 $\pm$ 0.007,0.629$\pm$ 0.007] | $5.79\%$ | — | — | [0.649 $\pm$ 0.006,0.661 $\pm$ 0.006] |  1.07%
V16 | 0.555 $\pm$ 0.004 | 15.39% | 0.562 $\pm$ 0.015 | 14.33% | 0.668 $\pm$ 0.006 |  1.83% | [0.549 $\pm$ 0.004,0.560 $\pm$ 0.004] | 16.31% | [0.548 $\pm$ 0.014,0.576 $\pm$ 0.014] | 16.46% | [0.660 $\pm$ 0.007,0.675 $\pm$ 0.007] |  2.90%
Avg Error | 8.23% |  | 14.33% |  |  1.07% |  | 9.14% |  | 16.46% |  |  2.14%
BlackHair with GT class probability $p^{*}_{0}$=0.668
R18 | 0.612 $\pm$ 0.005 | 8.38% | — | — | 0.659 $\pm$ 0.006 | 1.35% | [0.605 $\pm$ 0.005,0.620 $\pm$ 0.006] | 9.43% | — | — | [0.649 $\pm$ 0.004,0.670 $\pm$ 0.004] | 2.84%
R34 | 0.581 $\pm$ 0.006 | 13.02% | — | — | 0.662 $\pm$ 0.006 |  0.90% | [0.576 $\pm$ 0.005,0.586 $\pm$ 0.006] | 13.77% | — | — | [0.656 $\pm$ 0.005,0.669 $\pm$ 0.005] |  1.80%
MN2 | 0.596 $\pm$ 0.006 | 10.78% | — | — | 0.659 $\pm$ 0.005 |  1.35% | [0.591 $\pm$ 0.006,0.600 $\pm$ 0.007] | 11.50% | — | — | [0.652 $\pm$ 0.005,0.666$\pm$ 0.005] |  2.40%
V16 | 0.625 $\pm$ 0.006 | 6.44% | 0.608 $\pm$ 0.014 | 8.98% | 0.677 $\pm$ 0.005 |  1.35% | [0.620 $\pm$ 0.005,0.630 $\pm$ 0.006] | 7.19% | [0.590 $\pm$ 0.012,0.626 $\pm$ 0.013] | 11.68% | [0.670 $\pm$ 0.005,0.684 $\pm$ 0.006] |  2.40%
Avg Error | 9.66% |  | 8.98% |  |  1.24% |  | 10.47% |  | 11.68% |  |  2.36%

Table 12:  Comparing the point estimates and interval estimates of Baseline
and CLEAM in estimating the $\bm{p^{*}}$ of the Stable Diffusion Model [5]
with the GenData-SDM dataset. We use prompt input starting with "A photo of
with the face of" and ending with synonymous (Gender neutral) prompts. We
utilized CLIP as the sensitive attribute classifier for Gender, to obtain
$\bm{\hat{p}}$.

|  | Point Estimate | Interval Estimate
---|---|---|---
Prompt | GT | Baseline | CLEAM (Ours) | Baseline | CLEAM (Ours)
|  | $\mu_{\texttt{Base}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{CLEAM}}$ | $e_{\mu}(\downarrow)$ | $\rho_{\texttt{Base}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{CLEAM}}$ | $e_{\rho}(\downarrow)$
${\bm{\alpha}}$=[0.998,0.975], Avg. $\alpha$=0.987, CLIP –Gender
"A photo with the face of an individual" | 0.186 | 0.203 $\pm$ 0.011 | 9.14% | 0.187 $\pm$ 0.11 | 0.05% | [ 0.198 $\pm$ 0.10 , 0.208 $\pm$ 0.10 ] | 11.83% | [ 0.182 $\pm$ 0.10 , 0.192 $\pm$ 0.10 ] | 3.23%
"A photo with the face of a human being" | 0.262 | 0.277 $\pm$ 0.10 | 5.73% | 0.263 $\pm$ 0.10 | 0.38% | [ 0.270 $\pm$ 0.10 , 0.285 $\pm$ 0.10 ] | 8.78% | [ 0.255 $\pm$ 0.10 , 0.271 $\pm$ 0.10 ] | 3.44%
"A photo with the face of one person" | 0.226 | 0.241 $\pm$ 0.009 | 6.63% | 0.230 $\pm$ 0.08 | 1.77% | [ 0.232 $\pm$ 0.10 , 0.251 $\pm$ 0.10 ] | 11.06% | [ 0.220 $\pm$ 0.09 , 0.239 $\pm$ 0.09 ] | 5.75%
"A photo with the face of a person" | 0.548 | 0.556 $\pm$ 0.12 | 1.49% | 0.548 $\pm$ 0.11 | 0.00% | [ 0.545 $\pm$ 0.11 , 0.566 $\pm$ 0.11 ] | 3.28% | [ 0.537 $\pm$ 0.11 , 0.558 $\pm$ 0.11 ] | 2.01%
Average Error | 5.75% |  | 0.44% |  | 8.74% |  | 3.61%

###  D.2 Experimental Setup for Diversity[46]

In this section, we describe our experimental setup for Diversity [46], as
utilized in the main paper. Recall that as discussed by Kewsani et al. [46] a
VGG-16 [36] model pre-trained on ImageNet [58] is utilized as a feature
extractor. Then, this feature extractor is applied to both the unknown
(generator’s data) and the controlled dataset. Finally, the unknown sample’s
features are compared against the controlled one’s via a similarity algorithm
to compute diversity, $\delta$.

From our results in Fig. 7(a) (LHS) based on the pseudo-generator’s setup
(discussed in more details in Sec. D.3), we recognize that the original
implementation with VGG-16 trained on ImageNet works well on the Gender
sensitive attribute. This is seen by the close approximation made by the proxy
diversity score when compared against the GT diversity score evaluated with
Eqn. 15, as per [46].

$GT\,Diversity=p^{*}_{0}-p^{*}_{1}$ (15)

However, when evaluated on the harder BlackHair sensitive attribute, our
results in Fig. 7(a) (RHS) observed significant error between the GT Diversity
scores and the proxy Diversity scores. This error was especially prevalent in
the larger biases e.g. $p^{*}_{0}=0.9$. We theorized that this was due to the
differences between the domains of the feature extractor and the
generated/controlled images i.e. ImageNet versus CelebA/CelebA-HQ.

To verify this, we fine-tune the VGG-16 model on the CelebA dataset with the
respective sensitive attribute. Then we removed the last fully connected layer
of the classifier model, and utilise the 4096 feature vector for the diversity
measurement, as per [46]. Our results in Fig. 7(b) demonstrate significant
improvement on both Gender and BlackHair, based on the new improved VGG-16
model implementation. This thereby verifies our intuition that there exists a
mismatch of domains in the VGG-16 pretrained on ImageNet when utilized with
CelebA samples.

However, upon further experimentation, we recognize certain limitations still
exist in the Diversity measure when used on more ambiguous and harder
sensitive attribute e.g. Young and Attractive. Similar to before, we fine-
tuned the sensitive attribute classifier (feature extractor) which achieved
accuracies of $78.44\%$ and $84.41\%$ for Young and Attractive, respectively.
However even with this re-implementation, the diversity persistent to perform
poorly, as seen in Fig. 8.

Regardless, given the improvement seen on the BlackHair sensitive attribute,
we utilized our improved VGG-16 feature extractor in the main paper, in place
of the pre-trained VGG-16 (ImageNet).

(a) VGG-16 pre-trained on ImageNet

(b) VGG-16 pre-trained on ImageNet then fine-tuned on CelebA

Figure 7: Improvement in Diversity by fine-tuning the VGG-16, as a feature
extractor: (a) Diversity implementation by [46] with VGG-16 pre-trained on
ImageNet as the feature extractor testing on the pseudo-generator’s with
$p^{*}_{0}=\\{0.9,0.8,0.7,0.6,0.5\\}$ for sensitive attribute Gender(Left) and
BlackHair(Right). (b) We re-implemented VGG-16 and furter fine-tune it with
CelebA as the feature extractor. We observed improvement in predicting the GT
$\bm{p^{*}}$

(a) VGG-16(CelebA) on Attractive

(b) VGG-16(CelebA) on Young

Figure 8: Limitations Of Diversity algorithm. Our implementation of VGG-16
fine-tuned on CelebA w.r.t. sensitive attribute Attractive and Young. VGG-16
Classifier achieved an accuracy of $78.44\%$ and $84.1\%$ for sensitive
attribute Attractive and Young. However, the same VGG-16 performs poorly on
the diversity metric, demonstrating the limitations of the diversity
framework.

###  D.3  Measuring Varying Degrees of Bias

CLEAM for Measuring Varying Degrees of Bias. In previous experiments, we show
the performance of different methods in measuring the fairness of generators
and evaluating bias mitigation techniques. Another interesting analysis would
be to see how these methods fare with different bias, i.e. different
$\bm{p^{*}}$ values. A challenge of this analysis is that we cannot control
the training dynamics of either the GANs nor the Stable Diffusion Model to
obtain an exact value of $\bm{p^{*}}$. Thus, we introduce a new setup and use
a pseudo-generator instead of real GANs.

In this setup, we utilize the CelebA [47] and the AFHQ [48] dataset to
construct different modified datasets that follow different values of
$\bm{p^{*}}$ w.r.t. the sensitive attribute e.g. BlackHair attribute, when
$\bm{p^{*}}=\\{0.9,0.1\\}$, the modified dataset contains 4880 BlackHair and
542 Non-BlackHair samples. Then, a pseudo-generator with bias $\bm{p^{*}}$
works by random sampling from the corresponding datasets. Note that the
samples in the modified dataset are unseen to the sensitive attribute
classifier. For our experiment, we use different GT values,
$\bm{p^{*}}=\\{p_{0}^{*},p_{1}^{*}\\}$, where
$p_{0}^{*}\in\\{0.9,0.8,0.7,0.6,0.5\\}$, and $p_{1}^{*}=1-p_{0}^{*}$. For a
pseudo-generator, to calculate each value of $\bm{\hat{p}}$, a batch of $n$
samples is randomly drawn from the corresponding dataset and fed into the
$C_{\mathbf{u}}$ for classification. We utilize a ResNet-18 to evaluate our
pseudo-generator. The results in Tab. 13 for $p^{*}_{0}$ demonstrate that
CLEAM is effective for different degrees of bias, reducing the average error
($e_{\mu}$) of the Baseline from 1.43%$\rightarrow$0.27% and
6.23%$\rightarrow$0.49% for Gender and BlackHair on celebA respectively, and
3.52%$\rightarrow$0.75% for Cat/Dog on AFHQ. Additionally, note how
measurement error in Baseline and Diversity increases by increasing the data
bias, while CLEAM remains consistently low. See Sec. D.4 and D.5 for analysis
with more attributes and classifiers.

Table 13:  Comparing the point estimates and interval estimate of Baseline
[1], Diversity [46] and CLEAM in estimating different $\bm{p^{*}}$ of a
pseudo-generator, based on CelebA [47] and AFHQ [48], for sensitive attribute
Gender, BlackHair and Cat/Dog. The $\bm{\hat{p}}$ is computed with a ResNet-18
and the error rate is reported per Eqn.1 of the main manuscript

| Point Estimate | Interval Estimate
---|---|---
GT | Baseline | Diversity | CLEAM (Ours) | Baseline | Diversity | CLEAM (Ours)
| $\mu_{\texttt{Base}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{Div}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{CLEAM}}$ | $e_{\mu}(\downarrow)$ | $\rho_{\texttt{Base}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{Div}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{CLEAM}}$ | $e_{\rho}(\downarrow)$
${\bm{\alpha}}$=[0.976,0.979], Gender (CelebA)
$p^{*}_{0}$=0.9 | 0.880 | 2.22% | 0.950 | 5.55% | 0.899 | 0.11% | [0.876 , 0.884 ] | 2.67% | [0.913 , 0.986 ] | 9.56% | [0.895, 0.904] |  0.56%
$p^{*}_{0}$=0.8 | 0.783 | 2.10% | 0.785 | 1.88% | 0.798 |  0.25% | [ 0.778 , 0.788 ] | 2.75% | [0.762 , 0.809 ] | 4.75% | [0.794,0.803] |  0.75%
$p^{*}_{0}$=0.7 | 0.691 | 1.29% | 0.709 | 1.29% | 0.701 |  0.14% | [ 0.687 , 0.695 ] | 1.86% | [0.696 , 0.722 ] | 3.14% | [0.697, 0.707] |  0.10%
$p^{*}_{0}$=0.6 | 0.592 | 1.33% | 0.591 | 1.50% | 0.597 |  0.50% | [0.586 , 0.598 ] | 2.33% | [0.581 , 0.612 ] | 3.17% | [0.591,0.603] |  1.50%
$p^{*}_{0}$=0.5 | 0.501 |  0.20% | 0.481 | 3.80% | 0.502 | 0.40% | [0.495 , 0.507 ] |  1.40% | [0.473 , 0.490 ] | 5.40% | [0.497, 0.508] | 1.60%
Average Error: | 1.43% |  | 2.80% |  |  0.27% |  | $2.20\%$ |  | $5.20\%$ |  | $\mathbf{0.90}\%$
${\bm{\alpha}}$=[0.881,0.887], BlackHair (CelebA)
$p^{*}_{0}$=0.9 | 0.803 | 10.77% | 0.803 | 10.77% | 0.899 |  0.11% | [ 0.800 , 0.806 ] | 11.11% | $[0.791,0.815]$ | 12.11% | [0.893, 0.905] |  0.78%
$p^{*}_{0}$=0.8 | 0.723 | 9.63% | 0.699 | 12.63% | 0.796 |  0.50% | [0.719 , 0.727 ] | 10.13% | [0.686 , 0.713 ] | 14.25% | [0.790, 0.803] |  1.25%
$p^{*}_{0}$=0.7 | 0.654 | 6.57% | 0.661 | 5.57% | 0.705 |  0.71% | [ 0.648 , 0.660 ] | 7.43% | [ 0.643 , 0.68 ] | 8.14% | [0.698, 0.712] |  1.71%
$p^{*}_{0}$=0.6 | 0.575 | 4.17% | 0.609 | 1.50% | 0.602 |  0.33% | [ 0.564 , 0.586 ] | 6.00% | [0.604 , 0.614 ] | 2.30% | [0.599, 0.606] |  1.00%
$p^{*}_{0}$=0.5 | 0.500 |  0.00% | 0.521 | 4.20% | 0.504 | 0.8% | [ 0.495 , 0.505 ] |  1.00% | [0.506 , 0.536 ] | 7.20% | [0.497, 0.511] | 2.20%
Average Error: | 6.23% |  | 6.93% |  |  0.49% |  | 7.13% |  | 8.80% |  |  1.39%
${\bm{\alpha}}$=[0.953,0.0.990], Cat/Dog (AFHQ)
$p^{*}_{0}$=0.9 | 0.862 | 4.44% | 0.855 | 5.00% | 0.903 |  0.33% | [ 0.859 , 0.865 ] | 4.56% | [ 0.844 , 0.866 ] | 6.22% | [ 0.900 , 0.907 ] |  0.78%
$p^{*}_{0}$=0.8 | 0.766 | 4.25% | 0.774 | 3.25% | 0.802 |  0.25% | [ 0.762 , 0.771 ] | 4.75% | [ 0.765 , 0.784 ] | 4.38% | [ 0.797 , 0.807 ] |  0.88%
$p^{*}_{0}$=0.7 | 0.677 | 3.29% | 0.670 | 4.29% | 0.707 |  1.00% | [ 0.672 , 0.682 ] | 4.00% | [ 0.655, 0.686 ] | 6.43% | [ 0.701 , 0.712 ] |  1.71%
$p^{*}_{0}$=0.6 | 0.583 | 2.83% | 0.551 | 8.17% | 0.607 |  1.17% | [ 0.578 , 0.588 ] | 3.67% | [ 0.540, 0.562 ] | 10.00% | [ 0.602 , 0.613 ] |  2.17%
$p^{*}_{0}$=0.5 | 0.486 | 2.80% | 0.469 | 6.20% | 0.505 |  1.00% | [ 0.480 , 0.493 ] | 4.00% | [ 0.458, 0.480 ] | 8.40% | [ 0.498 , 0.511 ] |  2.20%
Average Error: | 3.52% |  | 5.38% |  |  0.75% |  | 4.20% |  | 7.09% |  |  1.55%

###  D.4 Measuring Varying Degrees of Bias with Additional Sensitive
Attributes

In Sec. D.3, we demonstrate CLEAM’s ability to improve accuracy in
approximating $\bm{p^{*}}$ for the sensitive attributes Gender and BlackHair.
In this section, we extend the experiment on CelebA dataset but with harder
(lower ${\bm{\alpha}}$) sensitive attributes i.e. Young, and Attractive. We
did not include Diversity in this study, due to its poor performance on harder
sensitive attribute, as discussed in D.2.

From our results in Tab. 14, both Young and Attractive classifiers have
relatively large errors ($e_{\mu_{\texttt{Base}}}$) in the baseline i.e. on
average 17.63% and 12.69%, respectively. Then utilizing CLEAM, even with the
harder sensitive attributes, we are able to significantly reduce these errors
to $0.68\%$ and $0.94\%$. See Sec. B for more details regarding the effect
that the different degrees of inaccuracies in ${\bm{\alpha}}$ have on the
Baseline error.

Table 14: Comparing point estimate and interval estimate of Baseline [1], and
proposed CLEAM measurement framework on a pseudo-generator with sensitive
attribute $\\{\texttt{Young},\texttt{Attractive}\\}$

| Point Estimate | Interval Estimate
---|---|---
GT | Baseline | Diversity | CLEAM (Ours) | Baseline | Diversity | CLEAM (Ours)
| $\mu_{\texttt{Base}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{Div}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{CLEAM}}$ |  $e_{\mu}(\downarrow)$ | $\rho_{\texttt{Base}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{Div}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{CLEAM}}$ | $e_{\rho}(\downarrow)$
${\bm{\alpha}}$=[0.749,0.852], Young
$p^{*}_{0}=0.9$ | 0.690 | 23.33% | — | — | 0.905 |  0.56% | [0.684,0.695] | 24.00% | — | — | [0.890,0.920] |  2.22%
$p^{*}_{0}=0.8$ | 0.630 | 21.25% | — | — | 0.804 |  0.50% | [0.625,0.635] | 21.88% | — | — | [0.795,0.813] |  1.63%
$p^{*}_{0}=0.7$ | 0.570 | 18.57% | — | — | 0.698 |  0.29% | [0.565,0.575] | 19.29% | — | — | [0.690,0.706] |  1.43%
$p^{*}_{0}=0.6$ | 0.510 | 15.00% | — | — | 0.595 |  0.83% | [0.505,0.515] | 15.83% | — | — | [0.590,0.600] |  1.67%
$p^{*}_{0}=0.5$ | 0.450 | 10.0% | — | — | 0.506 |  1.20% | [0.445,0.455] | 11.00% | — | — | [0.502,0.510] |  2.00%
Avg Error | 17.63% |  | —% |  |  0.68% |  | 18.40% |  | —% |  |  1.79%
${\bm{\alpha}}$=[0.780,0.807], Attractive
$p^{*}_{0}=0.9$ | 0.730 | 18.89% | — | — | 0.908 |  0.89% | [0.724,0.736] | 19.56% | — | — | [0.900,0.916] |  1.78%
$p^{*}_{0}=0.8$ | 0.670 | 16.25% | — | — | 0.804 |  0.50% | [0.665,0.675] | 16.88% | — | — | [0.795,0.813] |  1.63%
$p^{*}_{0}=0.7$ | 0.600 | 14.29% | — | — | 0.696 |  0.57% | [0.594,0.606] | 15.14% | — | — | [0.690,0.712] |  1.71%
$p^{*}_{0}=0.6$ | 0.540 | 10.00% | — | — | 0.592 |  1.33% | [0.534,0.546] | 11.00% | — | — | [0.580,0.604] |  3.33%
$p^{*}_{0}=0.5$ | 0.480 | 4.00% | — | — | 0.493 |  1.40% | [0.475,0.485] | 5.00% | — | — | [0.487,0.499] |  2.60%
Avg Error | 12.69% |  | —% |  |  0.94% |  | 13.52% |  | —% |  |  2.22%

###  D.5 Measuring Varying Degrees of Bias with Additional Sensitive Attribute
Classifier

In this section, we validate CLEAM’s versatility with different sensitive
attribute classifier architectures. In our setup, we utilise MobileNetV2 [35]
as in [7]. Then similar to Sec. D.3, we utilize a pseudo-generator with known
GT $\bm{p^{*}}$ for Gender and BlackHair sensitive attribute from the CelebA
[47] dataset, and Cat/Dog from the AFHQ [48] dataset, to evaluate CLEAM’s
effectiveness at determining bias.

As seen in our results in Tab.15, MobileNetV2 achieved reasonably high average
accuracy $\in$[0.889,0.983]. Then, when evaluating $\bm{p^{*}}$ of the pseudo-
generator we observed similar behavior to ResNet-18 discussed in sec. D.3. In
particular, we observed a significantly large $e_{\mu_{\texttt{Base}}}$ (for
the baseline) of $1.42\%$, $9.74\%$ and $2.81\%$ for the Gender, BlackHair and
Cat/Dog sensitive attribute, respectively. Whereas, CLEAM reported an
$e_{\mu_{\texttt{CLEAM}}}$ of $0.13\%$, $0.7\%$ and $0.62\%$, respectively.
The same trend can be observed in the IE. We thus demonstrate CLEAM’s
versatility and ability to be deployed as a post-processing method (without
retraining), on models of varying architecture.

Table 15:  Comparing the point estimates and interval estimate of Baseline
[1], Diversity [46] and proposed CLEAM measurement frameworks in estimating
different $\bm{p^{*}}$ of a pseudo-generator, based on the CelebA [47] and
AFHQ [48] dataset. The $\bm{\hat{p}}$ is computed with a MobileNetV2[35]
classifier and the error rate is reported using Eqn. 1 of the main manuscript.
We repeat this on Gender (CelebA), BlackHair (CelebA) and Cat/Dog(AFHQ)
sensitive attribute.

| Point Estimate | Interval Estimate
---|---|---
GT | Baseline | Diversity | CLEAM (Ours) | Baseline |  Diversity | CLEAM (Ours)
| $\mu_{\texttt{Base}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{Div}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{CLEAM}}$ | $e_{\mu}(\downarrow)$ | $\rho_{\texttt{Base}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{Div}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{CLEAM}}$ | $e_{\rho}(\downarrow)$
${\bm{\alpha}}$=[0.980,0.986], Gender (CelebA)
$p^{*}_{0}=0.9$ | 0.882 | 2.00% | 0.950 | 5.55% | 0.899 |  0.11% | [ 0.879 , 0.885 ] | 2.33% | [0.913 , 0.986 ] | 9.56% | [0.895,0.902] |  0.56%
$p^{*}_{0}=0.8$ | 0.786 | 1.75% | 0.785 | 1.88% | 0.800 | $\mathbf{0.00}\%$ | [ 0.782 , 0.790 ] | 2.25% | [0.762 , 0.809 ] | 4.75% | [0.794,0.804] |  0.75%
$p^{*}_{0}=0.7$ | 0.689 | 1.57% | 0.709 | 1.30% | 0.699 |  0.14% | [ 0.685, 0.693 ] | 2.14% | [0.696 , 0.722 ] | 3.14% | [0.694,0.704] |  0.86%
$p^{*}_{0}=0.6$ | 0.593 | 1.17% | 0.591 | 1.50% | 0.600 |  0.00% | [ 0.585 , 0.597 ] | 2.50% | [0.581 , 0.612 ] | 3.17% | [594,0.605] |  1.00%
$p^{*}_{0}=0.5$ | 0.497 | 0.60% | 0.481 | 3.80% | 0.502 |  0.40% | [ 0.491 , 0.502 ] | 1.80% | [0.473 , 0.490 ] | 5.40% | [495,0.507] |  1.40%
Avg Error | 1.42% |  | 2.81% |  |  0.13% |  | 2.20% |  | 5.20% |  |  0.91%
${\bm{\alpha}}$=[0.861,0.916], BlackHair (CelebA)
$p^{*}_{0}=0.9$ | 0.782 | 13.11% | 0.803 | 10.78% | 0.899 |  0.11% | [ 0.777 , 0.787 ] | 13.67% | [0.791 , 0.815 ] | 9.44% | [0.893,0.900] |  0.78%
$p^{*}_{0}=0.8$ | 0.705 | 11.88% | 0.699 | 12.63% | 0.800 |  0.00% | [ 0.699 , 0.710 ] | 12.63% | [0.686 , 0.713 ] | 14.25% | [0.793,0.807] | $\textbf{0.88}\%$
$p^{*}_{0}=0.7$ | 0.623 | 11.00% | 0.661 | 5.56% | 0.700 |  0.00% | [ 0.618 , 0.628 ] | 11.71% | [ 0.643 , 0.68 ] | 8.14% | [0.694,0.706] |  0.86%
$p^{*}_{0}=0.6$ | 0.550 | 8.33% | 0.609 | 1.50% | 0.600 |  0.00% | [ 0.544 , 0.556 ] | 9.33% | [0.604 , 0.614 ] | 2.33% | [0.593,0.608] |  1.17%
$p^{*}_{0}=0.5$ | 0.478 | 4.40% | 0.521 | 4.20% | 0.506 |  1.20% | [ 0.472 , 0.484 ] | 5.60% | [0.506 , 0.536 ] | 7.20% | [0.498,0.514] |  2.80%
Avg Error | 9.74% |  | 6.93% |  |  0.70% |  | $10.59\%$ |  | $8.27\%$ |  | $\mathbf{1.30}\%$
${\bm{\alpha}}$=[0.964,0.897], Cat/Dog (AFHQ)
$p^{*}_{0}=0.9$ | 0.875 | 2.77% | 0.880 | 3.26% | 0.897 |  0.34% | [ 0.872 , 0.878 ] | 3.07% | [0.871 , 0.890] | 3.25% | [ 0.894 , 0.900 ] |  0.68%
$p^{*}_{0}=0.8$ | 0.784 | 2.00% | 0.770 | 3.75% | 0.791 |  1.11% | [ 0.780 , 0.788 ] | 2.53% | [0.759 , 0.781 ] | 5.12% | [ 0.786 , 0.796 ] |  0.42%
$p^{*}_{0}=0.7$ | 0.704 | 0.62% | 0.692 | 1.08% | 0.698 |  0.20% | [ 0.700 , 0.708 ] | 1.19% | [ 0.684, 0.709 ] | 2.40% | [ 0.694 , 0.703 ] |  0.86%
$p^{*}_{0}=0.6$ | 0.617 | 2.78% | 0.611 | 1.83% | 0.597 |  0.54% | [ 0.611 , 0.622 ] | 2.78% | [0.602 , 0.620 ] | 3.42% | [ 0.591 , 0.603 ] |  1.58%
$p^{*}_{0}=0.5$ | 0.529 | 5.87% | 0.536 | 7.20% | 0.495 |  0.93% | [ 0.523 , 0.536 ] | 7.17% | [0.524 , 0.548 ] | 9.68% | [ 0.488 , 0.503 ] | $\mathbf{2.44}\%$
Avg Error | 2.81% |  | 3.42% |  |  0.62% |  | 3.35% | Avg Error | 4.77% |  |  1.20%

###  D.6 Measuring SOTA GANs and Diffusion Models with Additional Classifier

In this section, we further explore the utilization of CLIP as a sensitive
attribute classifier; more details on CLIP in Sec. E. Here, we follow the
setup in Sec. 5.1 of our main manuscript to measure the bias in GenData-
StyleGAN2 and GenData-StyleSwin w.r.t. Gender. Additionally, we evaluate a
publically available pre-trained Latent Diffusion Model (LDM) [59] on FFHQ
[3], where we acquire the GT $p^{*}$ w.r.t. Gender with the same procedure as
GenData.

Our results in Tab. 16 and 17 shows that the Baseline is able to achieve
reasonable accuracy in estimating the GT $\bm{p^{*}}$. This is because CLIP’s
accuracy is very high ($\alpha_{0}$=0.998) on the bias class ($p^{*}_{0}$) for
both StyleGAN2, StyleSwin and LDM resulting in less mis-classification from
occurring. Regardless, CLEAM is still able to further improve on the already
very accurate baseline, further reducing the error from
$e_{\mu_{\texttt{Base}}}\geq 0.91\%$, on StyleGAN2, StyleSwin and LDM to
$e_{\mu_{\texttt{CLEAM}}}\leq 0.47\%$. A similar trend can be observed in the
IE, where it is able to bound the GT $p^{*}_{0}$.

Table 16:  Comparing the point estimates and interval estimates of Baseline
[1] our CLEAM in estimating $\bm{p^{*}}$ of StyleGAN2 [3] and StyleSwin [4]
with the GenData datasets. We utilize SA classifier CLIP to classify sensitive
attribute Gender. The $p^{*}_{0}$ value of each GAN w.r.t. SA is determined by
manually hand-labeling the generated data. We repeat this for 5 experimental
runs and report the mean error rate, per Eqn. 1 of the main manuscript.

|  |  | Point Estimate | Interval Estimate
---|---|---|---|---
| ${\bm{\alpha}}=\\{\alpha_{0},\alpha_{1}\\}$ | Avg. ${\bm{\alpha}}$ | Baseline | Diversity | CLEAM (Ours) | Baseline | Diversity | CLEAM (Ours)
|  |  | $\mu_{\texttt{Base}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{Div}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{CLEAM}}$ | $e_{\mu}(\downarrow)$ | $\rho_{\texttt{Base}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{Div}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{CLEAM}}$ | $e_{\rho}(\downarrow)$
(A) StyleGAN2
Gender with GT class probability $p^{*}_{0}$=0.642
CLIP | {0.998, 0.975} | 0.987 | 0.653 | 1.71% | — | — | 0.645 | 0.47% | [0.649, 0.657] | 2.34% | — | — | [0.641, 0.649] | 1.09%
(B) StyleSwin
Gender with GT class probability $p^{*}_{0}$=0.656
CLIP | {0.998, 0.975} | 0.987 | 0.666 | 0.91% | — | — | 0.658 | 0.30% | [0.663,0.669] | 1.98% | — | — | [0.655,0.662] | 0.91%

Table 17:  Comparing the point estimates and interval estimates of Baseline
[1] our CLEAM in estimating $\bm{p^{*}}$ of a pretrained Latent Diffusion
Model[59] on the FFHQ dataset. We repeat this for 5 experimental runs and
report the mean error rate, per Eqn. 1 of the main manuscript.

|  |  | Point Estimate | Interval Estimate
---|---|---|---|---
| ${\bm{\alpha}}=\\{\alpha_{0},\alpha_{1}\\}$ | Avg. ${\bm{\alpha}}$ | Baseline | Diversity | CLEAM (Ours) | Baseline | Diversity | CLEAM (Ours)
|  |  | $\mu_{\texttt{Base}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{Div}}$ | $e_{\mu}(\downarrow)$ | $\mu_{\texttt{CLEAM}}$ | $e_{\mu}(\downarrow)$ | $\rho_{\texttt{Base}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{Div}}$ | $e_{\rho}(\downarrow)$ | $\rho_{\texttt{CLEAM}}$ | $e_{\rho}(\downarrow)$
Latent Diffusion Model
Gender with GT class probability $p^{*}_{0}$=0.570
CLIP | {0.998, 0.975} | 0.987 | 0.585 | 2.63% | — | — | 0.571 | 0.18% | [0.578, 0.593] | 4.04% | — | — | [0.564, 0.579] | 1.58%

###  D.7 Comparing Classifiers Accuracy on Validation Dataset vs Generated
Dataset

In our proposed CLEAM, we use ${\bm{\alpha}}$ pre-measured on the validation
dataset, denoted by ${\bm{\alpha}}_{val}$. In this section, we show that
${\bm{\alpha}}_{val}$ is a good approximate of the ${\bm{\alpha}}$ when
measured on the generated data, denoted by ${\bm{\alpha}}_{gen}$. Note that