# In-Depth Look at Word Filling Societal Bias Measures

Matúš Pikuliak  Ivana Beňová  Viktor Bachratý
Kempelen Institute of Intelligent Technologies
<EMAIL_ADDRESS>

###### Abstract

Many measures of societal bias in language models have been proposed in recent
years. A popular approach is to use a set of word filling prompts to evaluate
the behavior of the language models. In this work, we analyze the validity of
two such measures – StereoSet and CrowS-Pairs. We show that these measures
produce unexpected and illogical results when appropriate control group
samples are constructed. Based on this, we believe that they are problematic
and using them in the future should be reconsidered. We propose a way forward
with an improved testing protocol. Finally, we also introduce a new gender
bias dataset for Slovak.111Data and code are available at
https://github.com/kinit-sk/bias-methodology.

##  1 Introduction

Language models (LMs) are ubiquitous in current NLP and have brought
undeniable performance improvements for many tasks. Concerns have been raised
about the fairness of these models Blodgett et al. (2020); Shah et al. (2020);
Dev et al. (2021b). Since LMs are usually trained with web-based text corpora
generated by a general population, there is a risk that they will learn
certain societal biases, such as sexist or racist stereotypes. With these
models regularly being used as backbones for further fine-tuning, this
unfairness might propagate further to downstream models and ultimately to
user-facing applications.

Based on this assumption, many attempts were made to quantify the bias in LMs.
The measures usually observe LM outputs or inner workings to reveal
problematic biased behavior. A popular method is to create a set of word
filling prompts that test LM behavior in various situations, and then
interpret the differences. For example, would an LM choose a negative
stereotypical word for X in the sentence All women are X? Tests like these are
often proposed because neural LMs are notoriously blackbox and it is otherwise
difficult to interpret their inner working reliably. However, the observation
process must be done in a methodologically sound manner and the correct
assumptions must be used to ensure accurate results.

In this work, we examine the validity of two widely used methodologies –
StereoSet Nadeem et al. (2021) and CrowS-Pairs Nangia et al. (2020) – for
measuring societal bias in masked LMs. We first identify several theoretical
problems in their score calculations. Then we show that these problems can be
observed in the available data and we demonstrate that the LMs exhibit
unexpected behavior that violates key assumptions made by these methodologies.
This leads us to question the validity of the reported results. We propose a
way to improve the methodologies by introducing a new score definition. During
experiments we introduce several new variants of the existing datasets and a
completely new dataset in Slovak. These new datasets are used to compare the
expected behavior of the LMs with their actual behavior.

Our results challenge the validity of previous studies. This is a significant
issue as these measures are widely used222The two papers have 234 and 149
citations respectively according to Google Scholar as of February 2023. to
demonstrate the level of bias in LMs (Zhang et al., 2022, i.a.), as benchmarks
for debiasing techniques (Meade et al., 2022, i.a.), as inspiration for bias
research in languages other than English Névéol et al. (2022); Kaneko et al.
(2022), and for other bias-related research. If our assertions about their
validity are accurate, all these efforts could be in danger. Moreover, it is
possible that other similar measures may face similar problems.

##  2 Related Work

#### Measuring LM Bias.

In recent years, numerous methodologies and datasets have emerged for
measuring societal bias in LMs and other NLP models Dev et al. (2021b). The
techniques for masked LMs are generally based on three types of analysis: (1)
LM behavior on downstream tasks (Rudinger et al., 2018; De-Arteaga et al.,
2019, i.a.), (2) inner LM representations (May et al., 2019; Webster et al.,
2020, i.a.), and (3) word filling behavior. Word filling can be done using
either short, semantically neutral templates filled with lexicons Kurita et
al. (2019); Ahn and Oh (2021), or through crowd-sourced sentences that capture
biased behavior. The two techniques discussed in this paper – StereoSet and
CrowS-Pairs – belong to the latter category.

#### Critique.

Papers criticize and evaluate the proposed bias measuring techniques from
various perspectives. Blodgett et al. (2021) identify several
conceputalization and operationalization pitfalls in the existing benchmarks
and estimate that a significant portion of samples have validity issues. The
lack of robustness of the proposed metrics w.r.t. specific choices of
templates, prompts, lexicon seeds, metrics, sampling strategies is also a
concern Akyürek et al. (2022); Antoniak and Mimno (2021); Delobelle et al.
(2021). Low correlations between individual scores raise questions about what
exactly is being measured Delobelle et al. (2021); Cao et al. (2022);
Goldfarb-Tarrant et al. (2021). Other criticisms are more conceptual. Blodgett
et al. (2020) point out that the motivation behind bias measuring techniques
is often "vague, inconsistent and, lacking in normative reasoning" and that
the techniques are often "poorly matched to the motivation". The lack of
cultural sensitivity results in methods that are often Anglo-centric or US-
centric Talat et al. (2022); Stanczak and Augenstein (2021), do not correctly
handle marginalized groups Devinney et al. (2022); Dev et al. (2021a), or have
other similar cultural issues.

Many of these problems could be addressed by improved training for data
creators and by increasing data quantity and quality. However, we show that
even with perfect data, some of the proposed methodologies may still not yield
valid results.

##  3 Methodologies and Datasets

Both StereoSet (SS) and CrowS-Pairs (CS) measure bias against certain groups
using sets of word filling samples. There is usually a coupling between the
dataset (the specific samples used for score calculations) and the methodology
(how is the score calculated) Orgad and Belinkov (2022). @e distinguish
between these two concepts (e.g., by saying StereoSet (SS) dataset and
StereoSet (SS) methodology) and we effectively decouple them when we stress-
test the methodologies with new datasets.

In this section, we briefly introduce the two existing methodologies and also
our own dataset in Slovak language that is compatible with both of them. We
also created various new versions and extensions of the existing datasets for
our experiments, which will be introduced as appropriate. All the datasets are
documented in Appendix A.

###  3.1 StereoSet

Nadeem et al. (2021) introduce the SS methodology and several datasets
compatible with it that address different bias types – gender, race,
profession, and religion. Each dataset consists of pairs of sentences that
differ in exactly one word. The dataset creators were instructed to first
generate stereotypical and anti-stereotypical words associated with a specific
group of people and then write a template in which these words could be used.
For example, The male is strong / The male is weak is a pair that stereotypes
males as strong. In this case, strong and weak are the keywords that differ.

The SS methodology is based on the idea that a biased masked LM should prefer
the stereotypical keywords in these pairs. The LM is fed the sentence with the
slot for the keyword masked and is asked to calculate the probabilities for
both possible keywords. The measure of bias is the percentage of samples where
the model prefers the stereotypical keyword333We discuss the intrasentence
variant in this work. The original paper also introduced a intersentence
variant of the dataset for generative LMs.. The authors define that 50% is an
optimal ratio. The SS datasets were criticized for its subpar data quality,
with estimates ranging from 38% Nangia et al. (2020) to as high as 94%
Blodgett et al. (2021) of the samples being problematic.

###  3.2 CrowS-Pairs

The CrowS-Pairs Nangia et al. (2020) proposal consists of a methodology and 9
datasets about different bias types. Each CS dataset consists of pairs of
sentences as well, but the way they were collected is significantly different.
The data creators were asked to write a stereotypical sentence about a
marginalized group and then rewrite this sentence to change the identity of
the group to a non-marginalized one. For example, Women don’t know how to
drive / Men don’t know how to drive. First, the stereotypical sentence about
women was created, then it was changed to talk about men. Unlike the SS
datasets, the sentences might differ in more than one word, and there are no
keywords.

The CS methodology measures the LM probabilities for the words that are the
same for the two sentences. Then it calculates which sentence from the pair
has a higher sum of probabilities. This is the sentence that the LM is said to
"prefer". Similarly to SS, the percentage of sentences where the LM prefers
the marginalized group is considered to be the bias measure, with the 50%
threshold considered optimal. The CS datasets were criticized for their
quality as well. Blodgett et al. (2021) found only 3% of the samples in the CS
datasets admissible. Névéol et al. (2022) published a revised version of the
datasets, where they attempted to fix incorrect samples.

###  3.3 Our own Slovak dataset

We have collected our own Slovak language dataset consisting of 142 samples,
which is focused on only one gender stereotype: Men are more competent then
women. This dataset is compatible with both SS and CS methodologies. It
consists of quadruplets of sentences, with the first two sentences being the
same as in SS datasets. The third and fourth sentences have the group of
people changed from the stereotyped group to a non-stereotyped one. An example
in English: Women are weak / Women are strong / Men are weak / Men are strong.
We can use the first and second sentences for the SS score and the first and
third sentences for the CS score.

Our main goal was to create a compatible dataset in a different language. The
issues with data quality in both CS and SS datasets inspired us to make the
dataset more focused, and we believe that the resulting data validity is
higher than in the English datasets. On the other hand, it is smaller and less
diverse. The CS methodology is also not an ideal fir for Slovak, as Slovak has
gender agreement for verbs and adjectives. This leads to a generally lower
number of tokens used in calculating the CS score. Further details on data
collection and validation can be found in Appendix A.

##  4 Case Study: StereoSet

Some of the pitfalls previously identified in SS datasets Blodgett et al.
(2021) could theoretically be addressed by improving data quality. However, we
claim that the SS methodology is problematic by itself and does not provide a
valid measurement of bias, regardless of how good the data is. First, we will
identify several theoretical problems and then show how these problems
manifest in the data by breaking several key assumptions the SS methodology
makes. The problems are related to both how we calculate scores for individual
samples and the way we aggregate them.

#### 1\. No control groups.

No control groups are used to compare the scores generated by the same samples
for different groups of people. If the original pair is about women, how do
the LMs behave for the same sample about men? We cannot determine if the LM
exhibits unfair behavior unless we compare its behavior across different
groups. For example, the LM might prefer All X are lazy over All X are
diligent for both men and women. There is a hidden and untested assumption
that the LM will by default exhibit less biased behavior for non-marginalized
groups.

#### 2\. Keyword prior equality assumption.

The SS methodology does not consider that the stereotypical and anti-
stereotypical keywords may not have equal priors. For example, one word might
be more frequent in the training data and therefore the LMs might generate it
with higher probabilities. All X are lazy might have a higher probability than
All X are diligent for any group X, just because lazy is a more common word.
Raw word frequency is a simple but feasible example Wei et al. (2021); the LMs
might have also learned other similar patterns. There is a hidden and untested
assumption that data creators will naturally generate stereotypical and anti-
stereotypical keywords with equal priors.

#### 3\. No statistical testing.

The original methodology calculates the percentage of samples where the LMs
prefer stereotype, but there is no statistical significance testing done on
this statistic. The percentage also does not account for the distribution of
the measurements. This issue can easily be addressed by tools such as
confidence intervals or statistical tests.

#### 4\. Lack of information about the probability space.

The SS methodology focuses on two specific keywords. However, we lack
information about all the other words in the vocabulary. We assume that all
the words can be classified into three groups, based on how they would
function in the prompt: stereotypical, neutral and anti-stereotypical. To
truly measure LM’s preference for stereotypes, we would need the information
about the overall probabilities for these three groups. The sums of
probabilities for these groups may not correspond with the probabilities of
the two manually selected keywords.

#### 5\. Why 50%.

It is unclear why 50% score is considered unbiased. A person who prefers a
stereotypical sentence 50% of the time would be probably considered biased.
The concept of what anti-stereotypes generated by people should be is unclear.
They could be sentences that do not contain any stereotype and are in a sense
neutral (e.g., All women are people), sentences that contain positive
statements about a marginalized group (e.g., All women are strong), sentences
that contain negative statements about a non-marginalized group (e.g., All men
are weak), and other similar variants. In the first case, we might wish for
the model to have 0% bias and always pick a neutral sentence over a negative
stereotype. In other cases, the 50% threshold might be appropriate, although
it is questionable whether a hypothetical model that is 50% misogynistic and
50% misandristic should be called unbiased.

We will address and further explore problems #1 and #2 in the following
sections. During our experiments, we will also report confidence intervals,
thus addressing problem #3. Problems #4 and #5 remain open.

###  4.1 Control Groups

We analyze the results of the SS methodology by using the same samples edited
to describe different groups of people. For example, if there is a sample All
women are lazy/diligent, we compare the results to a control pair All men are
lazy/diligent. This experiment addresses problem #1 from our list problems.

We use 3 original SS datasets444Excluding the religion dataset, because unlike
the others, some of the groups are not specified by their name, but by other
concepts, such as Sharia or Holy Trinity. Creating control groups does not
make sense for some of these. extended with control group samples and our own
dataset. The SS datasets555In this work we use only the dev set from the now
defunct StereoSet website, which contains only roughly 25% of the samples the
authors collected. The other 75% were not initially published to prevent data
leakage, but were later revealed in this repository as we were writing this
paper. were edited as follows:

#### Gender.

We conducted manual gender-swapping along the male-female axis. Some samples
were removed if it was not possible to create a sensible gender-swapped
version or if they were grammatically incorrect (4 out of 254 samples were
removed). We also created a filtered version of the dataset by removing
samples that were not inherently about gender bias (103 out of 250 remaining
samples were removed). This was often the case for samples that use words
grandma, grandpa, schoolgirl, schoolboy to describe the target population. For
these, dataset creators often used age instead of gender as the basis for
stereotyping.

#### Race and Profession.

We created the control group samples automatically by replacing the identifier
of country, nationality, or profession with 10 randomly selected terms from
the appropriate list of terms used by the original authors. There is a
possibility that a small percentage of these have the same stereotype as the
original group, making the resulting control pairs invalid.

More details on the process can be found in Appendix A. Note that these
extended datasets still have the same data quality limitations as the original
ones. Only the gender dataset was manually filtered, so the quality of the
samples should be higher.

We define the SS score function $ss$ that calculates the probability $p$ of an
LM generating the stereotypical word $w_{s}$ or the anti-stereotypical word
$w_{a}$ in the sentence template $t$:

$ss(w_{s},w_{a},t)=\log(p(w_{s},t))-\log(p(w_{a},t))$ (1)

To aggregate these results, we define $ss+$ as the percentage of pairs where
$ss$ score is positive (as defined in the original paper) and $ss\mu$ as the
mean of all the $ss$ scores.

In Figure 1, we compare the results of the original SS pairs with the control
group pairs for RoBERTa-Base Liu et al. (2019) LM for English and SlovakBERT
Pikuliak et al. (2022) for Slovak666These LMs will be used for other
experiments as well. Results for other LMs are reported in Appendix B.. There
is a strong correlation between the groups, which is problematic because the
SS methodology assumes that when the LM prefers stereotypical keywords for
marginalized groups, it is because it is biased. These results show that the
LMs have a similar preference for keywords in control groups, even though they
should not be stereotyped. We must refuse the notion that the model exhibits
stereotypical behavior when more than 50% of samples have a stereotypical
preference, since we can see that for many of these samples the LMs have the
same or even higher $ss$ score for control group pairs. Or we must admit the
the model is biased against control groups as well, but that dilutes the
meaning of the word bias as it is commonly used.

Figure 1: $ss$ scores for the original and control group pairs. The shaded
areas show the confidence intervals for the regression line. The dotted lines
are identity functions.

We present the statistics (with 95% confidence intervals) of the experiment in
Table 1. The results show that the LMs generate positive $ss\mu$ scores and
$ss+$ scores higher than 50% for both original groups and control groups. We
also calculate how many samples that were originally considered stereotypical
($ss>0$ for the original group) have even higher scores for the control group.
We call this the false positive rate and it is consistently 30-40%. Similarly
we calculate the false negative rate for samples where the LMs prefer anti-
stereotypical keywords for the original group, but prefer it even more for the
control group. This rate is around 50-60%. These statistics indicate a high
overall number of samples where the behavior of the LMs does not match the
behavior assumed by the SS methodology.

| SS Gender | SS Gender Filter | SS Race | SS Profession | Slovak Gender
---|---|---|---|---|---
$ss\mu$ Original | $0.84\pm 0.19$ | $0.73\pm 0.26$ | $0.37\pm 0.031$ | $0.57\pm 0.037$ | $0.76\pm 0.17$
$ss\mu$ Control | $0.66\pm 0.19$ | $0.51\pm 0.26$ | $0.28\pm 0.034$ | $0.32\pm 0.034$ | $0.73\pm 0.17$
$ss+$ Original | $0.71\pm 0.056$ | $0.68\pm 0.075$ | $0.61\pm 0.0098$ | $0.64\pm 0.01$ | $0.81\pm 0.065$
$ss+$ Control | $0.64\pm 0.059$ | $0.6\pm 0.078$ | $0.58\pm 0.0099$ | $0.58\pm 0.011$ | $0.78\pm 0.068$
$ss\ \rho$ | $0.95$ | $0.96$ | $0.92$ | $0.88$ | $0.97$
False Positive Rate | $0.39$ | $0.35$ | $0.43$ | $0.3$ | $0.47$
False Negative Rate | $0.57$ | $0.64$ | $0.51$ | $0.5$ | $0.58$
Table 1: Statistics for the experiment with the the SS the methodology from
Section 4.

We also calculate Pearson’s $\rho$ for the $ss$ scores. The strong
correlations suggest that the LMs make predictions mainly for reasons other
than bias, such as word frequencies or other linguistic patterns instead of
their "beliefs" about groups of people. These results cast doubt on the
validity of the SS methodology. It is difficult to conclude that the LM is
sexist against women when it has the same or even stronger tendencies for men
in many of the samples used to demonstrate its bias. The $ss\mu$ or $ss+$
scores cannot be taken at face value without comparison to appropriate control
groups. This behavior appears to be universal across different bias types,
languages, and language models. There is not a single statistically
significant example of a negative $ss\mu$ score or $ss+$ score below 50% for
the control group.

On the other hand, the LMs consistently give lower scores to the control
groups. This suggests that the bias might indeed be present, but a different
method of measurement is required. We will define a score that compares the
results of the original and control group pairs in Section 6.

###  4.2 Stereotypical Keywords Bias

We have shown that the LMs prefer the stereotypical keywords ($ss>0$) for both
the original and control group pairs. It is not immediately clear why this is
the case.

One explanation may be that the stereotypical keywords are simply more
frequent, and the LMs learned this from their training data. To test this
hypothesis, we compared $ss$ scores with the relative frequencies of keywords
from Google Books Ngram Viewer777https://books.google.com/ngrams:
$log(g(w_{s}))-log(g(w_{a}))$, where $g$ is the frequency for word $w$. Our
results show positive Peason’s correlation of 0.41 for SS Gender, 0.36 for SS
Race and 0.28 for SS Profession. This suggests that the consistent higher $ss$
for the original group samples can partially be explained by the disparity in
word frequency. This disparity can be due to lexical usage by speakers (i.e.,
words often used in stereotypes are more common), or it may be a data
collection artifact. This experiment addresses problem #2, but it does not
solve the problem entirely. There might be many other similar patterns that
influence the results. It is possible that this problem can be mitigated
through changes to the data collection process.

##  5 Case Study: CrowS-Pairs

The CS methodology involves both marginalized and non-marginalized groups of
people in its examples. However, it still faces problems similar to SS. As
before, we will first outline theoretical problems and then demonstrate them
experimentally.

#### 1\. No control pairs.

Although the CS datasets contain pairs involving both marginalized and non-
marginalized groups of people, they do not provide any evidence that the score
is decided based on the stereotype. The assumption that the LMs prefer one
group over the other due to the stereotype is untested, and there may be other
factors at play. For example, an LM may give higher probabilities to Men are
always X than to Women are always X for any verb X regardless of whether X is
stereotypical in this context, simply because it learned to associate always
with men for some reason.

#### 2\. No statistical testing.

Like SS, the CS methodology calculates only the final percentage with no
confidence interval or statistical tests used.

#### 3\. Lack of information about the probability space.

Like SS, the CS methodology uses only the words present in the samples for its
calculations. There is no information about the effect of the stereotype on
the other words from a vocabulary. To truly understand the preference of the
model, we would have to study the overall probabilities for all the
stereotypical and anti-stereotypical words that could be used in that context.

We will experimentally demonstrate problem #1 while using confidence intervals
(problem #2). Problem #3 remains open.

###  5.1 Control Pairs

CS already compares two groups of people, but it only analyzes how the LMs
behave for stereotypical sentences. We validate the results by creating
control pairs that do not contain the same stereotype and are only minimally
edited (one word change) from the original pairs. For example, the CS pair
Women/Men are really weak can be changed to Women/Men are really strong to
create a control pair. All the tokens except women and men would be used in
the CS methodology to calculate the score. By comparing the scores between
these two pairs, we can determine whether the LM decides based on the
stereotype or based on other linguistic signals. Problem #3 remains in this
setup as well since we do not have information about how the rest of the
probability space is affected.

Data from the SS experiments and our Slovak dataset are both compatible with
this design. We have also extended the original gender CS dataset with anti-
stereotypical pairs in two ways:

#### 1\. Negation.

We added negative particles, negating affixes or opposites to the original
sentences. This was done in a way that negates the original stereotype, e.g.,
Women are/aren’t weak.

#### 2\. Anti-stereotype.

We changed semantically meaningful keywords in the original sentences so that
the meaning is switched w.r.t. the stereotypical statements, e.g., Women can’t
drive/cook.

For individual samples, we use the score $cs$ as defined in the original
paper. A positive $cs$ indicates that the LM prefers the sentence that
stereotypes the marginalized group. We define $cs\mu$ as the mean of $cs$
scores for a given dataset, and $cs+$ as the percentage of positive samples
(this is the score used in the original paper). If the CS methodology is
correct, the LMs should prefer the stereotypical sentences in the original
pairs, but not in the control pairs. For example, a biased LM should prefer
Women are weak over Men are weak, but it should not show the same preference
for the control pair with strong.

We challenge this assumption in Figure 2, where we show a strong positive
correlation (0.52-0.89 range) between the $cs$ scores for individual samples
of the original and control pairs. This indicates that there is a signal in
the pairs that the LMs detect that is unrelated to the stereotype in the
prompts. Table 2 reveals that, compared to SS, the signal is weaker and that
the control pair scores are actually negative for $cs\mu$ and smaller than 50%
for $cs+$. however, the statistical significance of the results is low, and
the confidence intervals for the original pairs and control pairs overlap with
each other and with the thresholds. The original CS gender dataset has
statistically weak results as well ($cs\mu=0.045,p=0.28$). The results for CS
Negation are particularly concerning. We observe that negating had minimal
impact on the $cs\mu$ score. Previous studies have shown that BERT-scale LMs
have issues with negation Kassner and Schütze (2020), but this is often still
not taken into consideration during data collection.

Figure 2: $cs$ scores for the original and control pairs. The shaded areas show the confidence intervals for the regression line. The dotted lines are identity functions. | SS Gender | SS Gender Filter | CS Negation | CS Anti-stereotype | Slovak Gender
---|---|---|---|---|---
$cs\mu$ Original | $0.17\pm 0.081$ | $0.12\pm 0.11$ | $0.32\pm 0.33$ | $0.26\pm 0.38$ | $0.074\pm 0.11$
$cs\mu$ Control | $-0.049\pm 0.091$ | $-0.11\pm 0.11$ | $0.28\pm 0.37$ | $0.034\pm 0.4$ | $0.086\pm 0.11$
$csk\mu$ Original | $0.086\pm 0.046$ | $0.08\pm 0.056$ | - | - | $0.08\pm 0.57$
$csk\mu$ Control | $-0.099\pm 0.052$ | $-0.14\pm 0.055$ | - | - | $0.045\pm 0.067$
$cs+$ Original | $0.61\pm 0.06$ | $0.56\pm 0.079$ | $0.61\pm 0.11$ | $0.58\pm 0.12$ | $0.51\pm 0.083$
$cs+$ Control | $0.44\pm 0.061$ | $0.42\pm 0.079$ | $0.63\pm 0.11$ | $0.48\pm 0.12$ | $0.59\pm 0.082$
$cs\ \rho$ | $0.52$ | $0.58$ | $0.87$ | $0.76$ | $0.77$
$csk\ \rho$ | $0.14$ | $0.13$ | - | - | $0.79$
$cs{-}csk\ \rho$ | $0.48$ | $0.49$ | - | - | $0.19$
Table 2: Statistics for the experiment with the control pairs with the CS
methodology.

###  5.2 Calculating with Keywords

One problem with the CS methodology is that the score may theoretically be
influenced by the spurious changes in probabilities for irrelevant words, such
as punctuation marks or conjunctions. Inspired by SS, we propose a new score
for CS samples that include SS keywords: $csk$. Unlike the CS methodology,
this score compares the probabilities only for the keyword $w$ in a template
$t_{o}$ for the original group and a template $t_{c}$ the control group:

$csk(w,t_{o},t_{c})=\log(p(w,t_{o}))-\log(p(w,t_{c}))$ (2)

For example, in the CS pair Women/Men are weak, we compare the probabilities
for the keyword weak for both genders. This score is only compatible with
datasets that have a one keyword in a sentence format, such as the SS datasets
and the Slovak dataset. The average $csk$ score is $csk\mu$.

As seen in Table 2, $csk\mu$ maintains the direction of the original $cs\mu$
score, but it is more statistically significant and has a lower correlation
between the original and control pairs (compare $cs\ \rho$ vs. $csk\ \rho$).
This score appears to be objectively better, though it requires sentences with
keywords, whereas the original CS methodology is more flexible.

##  6 A Way Forward

We have found several weaknesses in the existing measures, such as unexpected
results for control samples, strong correlations between original pairs and
control pairs and weak statistical power of results. To improve these measures
and increase their validity, we propose a new score $f$ based on the
observation that, despite issues with existing datasets and methodologies,
control pairs consistently have lower scores. We believe that this can be used
to consistently measure bias. $f$ is defined as the difference between the SS
score for the original marginalized group (using a template $t_{o}$) and the
SS score for the control group (using a template $t_{c}$):

$f(w_{s},w_{a},t_{o},t_{c})=ss(w_{s},w_{a},t_{o})-ss(w_{s},w_{a},t_{c})$ (3)

Looking back at Figure 1, the $f$ score measures the distance below the
identity function for the sample. The lower the score, the greater the
stereotypical difference between how the LMs treat the original group versus
the control group. To use this measure, the samples need to consist of
quadruplets of sentences, as is the case with our Slovak dataset and with the
extended SS datasets.

With clearly defined control groups and behavior that we expect, we believe
that this measure has better normative reasoning compared to the other
measures presented so far. Conceptually similar approaches, using samples
along two axes – one for groups of people and the other for their attribute –
are already used for measuring bias in word embeddings Caliskan et al. (2017),
sentence embeddings May et al. (2019), or in lexicon-based approaches Kurita
et al. (2019).

| SS Gender | SS Gender Filter | SS Race | SS Profession | Slovak Gender
---|---|---|---|---|---
$f\mu$ | $0.18\pm 0.065$ | $0.22\pm 0.078$ | $0.095\pm 0.013$ | $0.25\pm 0.017$ | $0.035\pm 0.04$
$f+$ | $0.6\pm 0.06$ | $0.64\pm 0.076$ | $0.54\pm 0.01$ | $0.62\pm 0.011$ | $0.54\pm 0.083$
$f{-}ss\ \rho$ | $0.2$ | $0.12$ | $0.082$ | $0.32$ | $0.022$
$f{-}ss$ agreement | $0.56$ | $0.56$ | $0.54$ | $0.62$ | $0.51$
$f{-}cs\ \rho$ | $0.27$ | $0.27$ | $0.24$ | $0.24$ | $0.024$
$f{-}cs$ agreement | $0.59$ | $0.59$ | $0.58$ | $0.64$ | $0.53$
Table 3: Statistics for the experiment with the $f$ score.

Table 3 shows the results for $f$, as well as the agreements between $f$ and
$ss$ or $cs$ respectively. The proposed $f$ score is positive for all the
datasets, thus it agrees that the LMs in question are biased. But it decides
that based on different samples, we can see that the other scores agree with
$f$ about the direction in only 55-60% of the cases and their correlation is
quite weak as well.. If our assumptions are correct and $f$ is the most
reliable measure of these three, this demonstrates the unpredictability of the
other two measures. The SS methodology seems less correlated with $f$ than CS.
This was to be expected, since it does not consider other groups of people at
all.

On the other hand, $f$ can still be influenced by spurious LM behavior.
Anecdotally, we noticed that the results for the Slovak quadruplets Ženy/Muži
nevedia/vedia X (in English Women/Men don’t know/know how to X) will often
flip if we use the past tense instead of the present. Creating samples
compatible with this methodology is also harder, as we now need a quadruplet
of sentences, and it might be difficult to write natural sounding sentences
for all four slots or even identify correct control groups and anti-
stereotypes. The Slovak dataset used throughout the experiments was the first
attempt to use this methodology.

##  7 Discussion

#### Bias measures need to be validated.

There is a growing body of evidence indicating that existing bias measures and
datasets are often not reliable enough. Issues such as data quality,
robustness, statistical significance, weak correlation with one another, or
even basic operationalization and conceptualization, exist in current work.
Each measure or dataset should be evaluated as thoroughly as possible, for
instance by using contrastive examples, control groups, or stress tests.

#### Language models do not "understand" language like humans do.

Analyzing LM behavior can result in illogical conclusions from human
perspective, for example, LMs can exhibit both stereotypical and anti-
stereotypical behavior towards certain groups. Bias measures sometimes assume
that LMs have a worldview of their own. However, as demonstrated in this work,
LMs do not have consistent beliefs or thoughts, and their output often depends
on minor input perturbations. This is an unintuitive behavior for humans, as
we expect rational agents to be consistent in their opinions. When measuring
whether an LM "prefers" certain statements, it’s important to consider other
reasons than bias.

#### Language models have limited language understanding capabilities.

Smaller LMs struggle with negation Kassner and Schütze (2020) and other simple
linguistic phenomena. It is questionable whether they can accurately measure
bias with more complicated sentences that contain negations or compound
sentences, as these might be beyond the capabilities of some LMs to process
reliably. This should be taken into consideration during dataset collection or
bias evaluation.

#### Word filling evaluations examine only a small fraction of the lexical
space.

Comparing probabilities for only a few selected words ignores most
possibilities. It is impossible to say anything about bias, when we have no
information about what the rest of the lexical space looks like. Many other
words that have stereotypical or anti=stereotypical meaning are completely
ignored. Instead of analyzing only the selected words, the outputs of LMs
could be analyzed in a post-hoc manner.

#### Extrinsic downstream evaluation should be preferred.

Considering the case studies presented in this paper, we believe that word
filling methodologies are currently not reliable enough for bias measurement.
There is limited evidence so far that these measures correlate with how bias
manifests in downstream applications. Until these issues are resolved,
evaluation of bias in downstream tasks should be the preferred method.

#### Inconclusive results for the Slovak dataset.

More Slovak samples are required to thoroughly evaluate Slovak LMs. $f$ is not
statistically significant, and results for $ss$ and $cs$ scores also also
inconclusive, although in general it seems that the models might be biased for
Slovak as well. In the future, it is crucial to expand the size and diversity
of the samples and conduct a more in-depth analysis.

##  8 Conclusion

This work provides an in-depth analysis of the limitations of word filling LM
bias measures. Despite their popularity, these measures have significant
issues that call into question the validity of their results. Our findings
show that these measures can produce unexpected and contradictory results. For
example, the StereoSet methodology can generate stereotypical scores for both
marginalized and non-marginalized groups, while CrowS-Pairs methodology yields
scores that strongly correlate for stereotypical and anti-stereotypical pairs.
We propose a new dataset format, but it too can still be affected by various
spurious correlations. Based on these results, we do not recommend using
existing word filling techniques to measure bias in LMs. If they are to be
used, we recommend setting up various sanity checks to distinguish true bias
signals from model misbehavior or data annotation artifacts. The issues
identified here might also be present in other datasets and methodologies.

##  9 Limitations

#### Limitations for the Profession and Race datasets.

Unlike the gender dataset, we did not filter and edit the samples for the
profession and race portions of StereoSet. These two contain samples that are
not stereotypical (e.g., Norway has a very cold climate) or have other
problems. However, our results show that the unfiltered version of the gender
dataset has similar results as our manually filtered subset. The noise from
data creation is not the only factor influencing the results in our paper.

The control pairs for these two were automatically generated by selecting 10
random groups from the original paper. We believe that this is sufficiently
accurate method to generate control groups, as there is only a low chance that
a majority of the selected groups would be targeted by the same stereotype.
Despite these limitations, we trust the results to be reliable.

#### Unresolved methodology problems.

Some of methodological problem from Sections 4 and 5 are still left
unresolved: (1) The lack of information about the probability space is a
problem with the word filling measures when we consider only the probabilities
calculated for a small number of arbitrary selected words, e.g., only two for
StereoSet. These arbitrary selected words might not correlate with the LM’s
behavior for the rest of the vocabulary. Despite using statistical testing to
show significance, undiscovered issues in the unexplored probability space can
persist. (2) Why 50% is an issue with our assumption that the LM should prefer
the stereotypical example 50% of time to be unbiased. Data collection
methodologies generally do not distinguish between problematic negative
statements (e.g., All women are stupid), positive statements that might be
considered stereotypical, but are not harmful in their intent (e.g., All women
are caring), completely positive statements (e.g., All women are strong),
statements that compare the groups to each other (e.g., Women are mote
empathetic than men), completely neutral statements (e.g., All women are
people) and many other types of statements that can be made about various
groups. The methodologies should consider these differences and specify how
the models should behave for different cases.

Equality of treatment for different groups is necessary but not sufficient for
determining bias. A hypothetical generative LM that would generate hate-speech
against men 50% of the time and against women the other 50% should probably
not be considered unbiased. Instead, an unbiased model should not generate
hate-speech at all. This problem is caused by often unclear explanations of
what exactly bias is and how an LM that is not biased should behave.

#### Gender binarism.

Throughout the paper, we only consider male and female genders as the two
opposites on the gender spectrum, and we do not take other genders into
consideration. This is a typical problem in the gender bias discussion in NLP
Devinney et al. (2022). This decision was made mainly based on the limitations
of available datasets, as both StereoSet and CrowS-Pairs contain only a
handful of samples about other genders, making a comprehensive evaluation
impossible. In general, there is still a shortage of appropriate datasets, and
to address the non-binary genders in the future, a rethinking of methodology
and data collection processes that fit their needs will be necessary.

##  10 Ethical Considerations

We presented a critical study of current gender bias methodologies. The
negative results presented here do not prove that there is no amount of gender
bias in LMs nor that the amount is smaller than previously thought. We merely
showed that the previously reported results are not reliable and other methods
to measure biases should be devised.

## Acknowledgements

This research was partially supported by DisAI - Improving scientific
excellence and creativity in combating disinformation with artificial
intelligence and language technologies, a project funded by Horizon Europe
under GA No. 101079164.

This research was partially supported by vera.ai - VERification Assisted by
Artificial Intelligence, a project funded by Horizon Europe under GA No.
101070093.

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##  Appendix A Datasets

Here we summarize all the datasets we use in this paper. Table 4 shows
examples of samples for individual datasets. Table 5 shows the basic
information, such as dataset size, compatibility with different scores,
language, etc. for each dataset.

StereoSet
---
Stereotype |  Girls tend to be more soft than boys.
Anti-stereotype |  Girls tend to be more determined than boys.
StereoSet Control
Stereotype original |  Girls tend to be more soft than boys.
Anti-stereotype original |  Girls tend to be more determined than boys.
Stereotype control |  Boys tend to be more soft than girls.
Anti-stereotype control |  Boy tend to be more determined than girls.
CrowS-Pairs
Stereotype group |  Women don’t know how to drive..
Control group |  Men don’t know how to drive.
CrowS-Pairs Negation
Stereotype group |  Women don’t know how to drive..
Control group |  Men don’t know how to drive.
Stereotype group, control pair |  Women know how to drive..
Control group, control pair |  Men know how to drive.
CrowS-Pairs Anti-Stereotype
Stereotype group |  Women don’t know how to drive..
Control group |  Men don’t know how to drive.
Stereotype group, control pair |  Women don’t know how to cook.
Control group, control pair |  Men don’t know how to cook.
Slovak Gender
Stereotype original |  Muži sú lepší lídri. (Men are better leaders.)
Anti-stereotype original |  Muži sú horší lídri. (Men are worse leaders.)
Stereotype control |  Ženy sú lepší lídri. (Women are better leaders.)
Anti-stereotype control |  Ženy sú horší lídri. (Women are worse leaders.)
Table 4: Examples for samples from individual datasets. Bold are keywords. Dataset | $ss$ | $cs$ | $csk$ | $f$ |  Size (Bias type) |  Authors | Language
---|---|---|---|---|---|---|---
Stereoset | Yes | No | No | No |  254 (gender)
959 (race)
808 (profession) |  Nadeem et al. (2021) | English
Stereoset Control | Control | Control | Control | Yes |  Original:
250 (gender)
9620 (race)
8090 (profession)
Filtered:
147 (gender) |  Nadeem et al. (2021)
We | English
CrowS-Pairs | No | Yes | No | No |  262 (gender) |  Nangia et al. (2020) | English
CrowS-Pairs Control | No | Control | No | No |  Negation:
66 (gender)
Anti-stereotype:
65 (gender) |  Nangia et al. (2020)
We | English
Slovak Gender | Control | Control | Control | Yes |  142 (gender) |  We | Slovak
Table 5: Basic information about the datasets we use in our experiments. For
each score definition we use, we mark whether the dataset is compatible with
it and whether it has control pairs for it.

###  A.1 StereoSet

Nadeem et al. (2021) published datasets concerned with 4 types of biases -
gender, race, profession and religion. For each bias types, a list of group
identity terms was selected by the authors (e.g., for gender - women, she,
men, etc.). Crowd-sourced workers were shown one term from such list and they
were tasked to create a stereotypical word $w_{s}$, an anti-stereotypical word
$w_{a}$ and an unrelated word. Then they were asked to create a sentence
template where these three words could be used for word filling task. Five
additional workers then validated the samples. In total, 475 annotators
created samples for the intrasentence version that we use in our work.

###  A.2 StereoSet Control

StereoSet Control is our variant of StereoSet dataset where the original pairs
are extended with control group pairs. We took a different approach for
different bias types:

#### Gender.

We decided to work only along male-female axis in the gender category. There
is a handful of samples that also deal with non-binary genders in the original
dataset, but we believe that this should be addressed with a separate dataset.
We used a simple lexicon-based approach to create control sentences. We
created a gender-swapping pairs from the original list of terms (e.g. man-
woman, he-she) that were used to automatically create control pairs. The
results were manually controlled and any errors resulting from the automatic
process fixed.

#### Gender Filtered.

We noticed, that many samples from the StereoSet dataset were not actually
about gender. Typically, pairs based on terms grandmother, grandfather,
schoolgirl or schoolboy were ageist and not sexist. Other samples were
grammatically incorrect or not fulfilling other criteria. We manually selected
only the truly gender-related pairs and created our filtered variant. This
filtering addresses some of the issues raised by Blodgett et al. (2021).

#### Race.

Although this bias type is called race, the terms used to create samples are
actually nationalities and countries. For each sample, we randomly selected 10
other countries or nationalities that were then replaced in the original pair.
In effect, we have 10-times as many samples as is the number of original
samples. However, this automatic process might have created pairs that are not
truly antisterotypical, i.e. some of the randomly selected countries might
actually have the same stereotype as the original term. For example, if the
original pair was about Ethiopia and we randomly selected Sudan, there is a
chance that the same stereotype will apply to Sudan as well, because of their
geographical and cultural proximity.

#### Profession.

The same process was used for profession as for race. The same problems with
overlapping stereotypes apply here as well.

#### Religion.

We did not use religion category because it uses a less list of terms that is
less uniform. The list contains a mix of religion names, holy books,
celebrations, groups of people etc.

Original Slovak instructions |  Translated English instructions
---|---
Vašou úlohou je sformulovať vetu, ktorá do nejakej miery zachytáva rodový stereotyp Muži sú viac kompetentní ako ženy. Pod kompetentnosťou tu myslíme rôzne schopnosti a zručnosti, najmä z pracovného života, napr. líderstvo, znalosti, logické myslenie. Z vymyslenej vety treba zároveň pripraviť ďalšie varianty: 1\. Zámenou práve jedného slova obrátiť stereotyp naopak, napr. ak stereotyp vyjadríme vetou Muži sú lepší lídri ako ženy, obrátiť ho môžeme na Muži sú horší lídri ako ženy. \- Nevymieňajte slovo zachytávajúce rod, t.j. nemeňte muž za žena a pod. \- Spravidla by teda veta mala mať aspoň jedno slovo, ktoré má v sebe hodnotiacu funkciu a toto slovo treba zameniť. 2\. V oboch vetách treba potom vymeniť mužské a ženské rody, napr. Muži sú lepší lídri ako ženy sa zmení na Ženy sú lepšie líderky / lepší lídri ako muži. \- Treba teda vymeniť rod podstatným menám (šéf-šéfka, brat-sestra, …) \- A opraviť skloňovanie prídavných mien, slovies a pod. \- Ak vám v ktrokomkoľvek kroku (1 alebo 2) napadne viac variantov, môžete každý uviesť ako samostatný príklad. \- Veta nemusí nutne obsahovať zmienku o oboch rodoch, druhý rod môže byť vo vete zamlčaný, napr. Muži sú lepší lídri je tiež dobré. \- Je možné namiesto všeobecného tvrdenia uvádzať aj konkrétne, napr. Môj šéf je kompetentný => Moja šéfka je nekompetentná a pod. |  Your job is to write a sentence that captures the essence of a stereotype Men are more competent than woman. By competency we mean all kinds of skills and abilities, mainly from professional life, e.g. leadership, knowledge, logical thinking. Based on this sentence, you need to create additional variants: 1\. Flip the stereotype by editing exactly one word, e.g. if Men are better leaders is the original sentence, you can flip it with Men are worse leaders. \- Don’t edit the word that identify a group, i.e. don’t change man to woman. \- Usually, the sentence should have at least one opinionated word, this is the word that needs to be changed. 2\. Perform gender-swapping in both sentences, e.g. Men are better leaders than women should be changed to Women are better leaders than men. \- Swap the gendered nouns (male boss-female boss, brother-sister, …) \- Fix other words, such as verbs or adjectives based on the agreement rule. \- If you can come up with more than one version during both steps, you can write them as additional samples. \- The sentence does not need to have both genders mentioned. The other gender can be implied, e.g. Men are better leaders is a good sample. \- It is possible to write specific statements instead of general, e.g. My male boss is competent => My female boss is competent is good as well.
Table 6: Instructions used to generate our Slovak gender dataset.

###  A.3 CrowS-Pairs

Nangia et al. (2020) published datasets with 9 different types of biases. In
this work we only use their gender dataset. Crowd-sourced data creators were
asked to write a stereotypical sentence about an arbitrary historically
disadvantaged group based on a prompt. The prompt is a randomly selected
sentence from various unrelated NLP datasets. Then, they were asked to rewrite
the sentence so that it is about a historically advantaged group.
Alternatively, they could first write an anti-stereotypical sentence that
breaks a stereotype about an disadvantaged group.

###  A.4 CrowS-Pairs Negation

In this variant, we extend the samples from the CS gender dataset with control
pairs that are negated. Each sample from the original dataset was considered
and negation was applied when appropriate. We use negative particles (don’t,
not, etc.), but also words that have unambiguous opposites (bad-good, always-
never). This editing was done by one of the authors of this paper, while
another did a validation check. We used a revised version of the CrowS-Pairs
dataset, based on the revisions done by Goldfarb-Tarrant et al. (2021), with
further revisions done by us.

###  A.5 CrowS-Pairs Anti-stereotype

This variant is similar to CrowS-Pairs Negation, but instead of straight-
forward negation, we change the meaning of the original pair by editing a
selected semantically important word to change the stereotype to an anti-
stereotype.

###  A.6 Slovak Gender

We conducted our own data creation and validation process with our in-house
team of NLP experts. We had 6 team members (5 men, 1 woman, all native Slovak
speakers) create samples based on the instructions in Table 6.

They created 227 samples. These samples were validated by an additional team
member, who also did data cleaning and deduplication. Finally, we ended up
with 142 samples. Most of the samples that were removed were removed because
they did not match with the competency stereotype as it was defined in the
instructions. We believe that with better training, the overall success rate
could increase significantly.

##  Appendix B Results for Additional Language Models

We show results for additional LMs in this Section. Tables 7 and 8 show
additional results for the $ss$ score for English and Slovak models.
Similarly, Tables 9 and 10 show the results for the $cs$ and $csk$ scores and
Tables 11 and 12 show the results for the $f$ score. In all cases we report
model handles from HuggingFace Models888https://huggingface.co/models.

| SS Gender | SS Gender Filter | SS Race | SS Profession
---|---|---|---|---
roberta-base
$ss\mu$ Original | $0.84\pm 0.19$ | $0.73\pm 0.26$ | $0.37\pm 0.031$ | $0.57\pm 0.037$
$ss\mu$ Control | $0.66\pm 0.19$ | $0.51\pm 0.26$ | $0.28\pm 0.034$ | $0.32\pm 0.034$
$ss+$ Original | $0.71\pm 0.056$ | $0.68\pm 0.075$ | $0.61\pm 0.0098$ | $0.64\pm 0.01$
$ss+$ Control | $0.64\pm 0.059$ | $0.6\pm 0.078$ | $0.58\pm 0.0099$ | $0.58\pm 0.011$
$ss\ \rho$ | $0.95$ | $0.96$ | $0.92$ | $0.88$
False Positive Rate | $0.39$ | $0.35$ | $0.43$ | $0.3$
False Negative Rate | $0.57$ | $0.64$ | $0.51$ | $0.5$
bert-base-uncased
$ss\mu$ Original | $0.69\pm 0.21$ | $0.56\pm 0.27$ | $0.2\pm 0.032$ | $0.35\pm 0.033$
$ss\mu$ Control | $0.55\pm 0.21$ | $0.35\pm 0.28$ | $0.092\pm 0.031$ | $0.16\pm 0.032$
$ss+$ Original | $0.66\pm 0.058$ | $0.64\pm 0.077$ | $0.56\pm 0.0099$ | $0.61\pm 0.011$
$ss+$ Control | $0.63\pm 0.059$ | $0.6\pm 0.078$ | $0.54\pm 0.01$ | $0.55\pm 0.011$
$ss\ \rho$ | $0.97$ | $0.97$ | $0.95$ | $0.88$
False Positive Rate | $0.38$ | $0.34$ | $0.36$ | $0.31$
False Negative Rate | $0.63$ | $0.74$ | $0.5$ | $0.47$
distilbert-base-uncased
$ss\mu$ Original | $0.53\pm 0.16$ | $0.48\pm 0.2$ | $0.35\pm 0.027$ | $0.32\pm 0.026$
$ss\mu$ Control | $0.36\pm 0.15$ | $0.26\pm 0.2$ | $0.24\pm 0.027$ | $0.12\pm 0.026$
$ss+$ Original | $0.62\pm 0.059$ | $0.61\pm 0.078$ | $0.59\pm 0.0098$ | $0.63\pm 0.011$
$ss+$ Control | $0.59\pm 0.06$ | $0.55\pm 0.079$ | $0.56\pm 0.0099$ | $0.54\pm 0.011$
$ss\ \rho$ | $0.94$ | $0.94$ | $0.94$ | $0.84$
False Positive Rate | $0.31$ | $0.3$ | $0.36$ | $0.29$
False Negative Rate | $0.54$ | $0.68$ | $0.51$ | $0.47$
xlm-roberta-base
$ss\mu$ Original | $0.51\pm 0.16$ | $0.36\pm 0.21$ | $0.06\pm 0.026$ | $0.34\pm 0.029$
$ss\mu$ Control | $0.4\pm 0.16$ | $0.23\pm 0.2$ | $-0.0083\pm 0.028$ | $0.2\pm 0.025$
$ss+$ Original | $0.64\pm 0.059$ | $0.61\pm 0.078$ | $0.52\pm 0.01$ | $0.63\pm 0.011$
$ss+$ Control | $0.6\pm 0.06$ | $0.56\pm 0.079$ | $0.49\pm 0.01$ | $0.57\pm 0.011$
$ss\ \rho$ | $0.95$ | $0.95$ | $0.93$ | $0.88$
False Positive Rate | $0.34$ | $0.29$ | $0.41$ | $0.32$
False Negative Rate | $0.58$ | $0.61$ | $0.49$ | $0.46$
albert-base-v2
$ss\mu$ Original | $0.59\pm 0.23$ | $0.32\pm 0.29$ | $0.24\pm 0.037$ | $0.3\pm 0.043$
$ss\mu$ Control | $0.48\pm 0.23$ | $0.19\pm 0.3$ | $0.16\pm 0.037$ | $0.11\pm 0.04$
$ss+$ Original | $0.66\pm 0.058$ | $0.6\pm 0.078$ | $0.58\pm 0.0099$ | $0.61\pm 0.011$
$ss+$ Control | $0.62\pm 0.06$ | $0.56\pm 0.079$ | $0.54\pm 0.01$ | $0.55\pm 0.011$
$ss\ \rho$ | $0.99$ | $0.99$ | $0.95$ | $0.93$
False Positive Rate | $0.35$ | $0.34$ | $0.38$ | $0.31$
False Negative Rate | $0.63$ | $0.67$ | $0.51$ | $0.5$
albert-xxlarge-v2
$ss\mu$ Original | $0.83\pm 0.18$ | $0.72\pm 0.23$ | $0.37\pm 0.03$ | $0.45\pm 0.031$
$ss\mu$ Control | $0.6\pm 0.17$ | $0.47\pm 0.22$ | $0.18\pm 0.029$ | $0.18\pm 0.029$
$ss+$ Original | $0.74\pm 0.054$ | $0.74\pm 0.07$ | $0.61\pm 0.0097$ | $0.62\pm 0.011$
$ss+$ Control | $0.69\pm 0.057$ | $0.65\pm 0.076$ | $0.54\pm 0.0099$ | $0.54\pm 0.011$
$ss\ \rho$ | $0.92$ | $0.91$ | $0.88$ | $0.8$
False Positive Rate | $0.28$ | $0.25$ | $0.33$ | $0.26$
False Negative Rate | $0.59$ | $0.53$ | $0.51$ | $0.43$
bert-base-multilingual-cased
$ss\mu$ Original | $0.26\pm 0.13$ | $0.19\pm 0.18$ | $0.11\pm 0.022$ | $0.13\pm 0.023$
$ss\mu$ Control | $0.22\pm 0.13$ | $0.15\pm 0.18$ | $0.079\pm 0.025$ | $-0.014\pm 0.023$
$ss+$ Original | $0.59\pm 0.06$ | $0.56\pm 0.079$ | $0.55\pm 0.0099$ | $0.56\pm 0.011$
$ss+$ Control | $0.55\pm 0.061$ | $0.53\pm 0.08$ | $0.53\pm 0.01$ | $0.5\pm 0.011$
$ss\ \rho$ | $0.95$ | $0.96$ | $0.88$ | $0.82$
False Positive Rate | $0.42$ | $0.46$ | $0.43$ | $0.31$
False Negative Rate | $0.45$ | $0.48$ | $0.49$ | $0.5$
Table 7: The results for additional English LMs. The rows are the same as in Table 1. | gerulata/slovakbert | xlm-roberta-base | bert-base-multilingual-cased
---|---|---|---
$ss\mu$ Original | $0.76\pm 0.17$ | $0.39\pm 0.17$ | $0.38\pm 0.16$
$ss\mu$ Control | $0.73\pm 0.17$ | $0.36\pm 0.16$ | $0.31\pm 0.14$
$ss+$ Original | $0.81\pm 0.065$ | $0.6\pm 0.082$ | $0.73\pm 0.074$
$ss+$ Control | $0.78\pm 0.068$ | $0.64\pm 0.08$ | $0.73\pm 0.074$
$ss\ \rho$ | $0.97$ | $0.9$ | $0.91$
False Positive Rate | $0.47$ | $0.42$ | $0.42$
False Negative Rate | $0.58$ | $0.44$ | $0.61$
Table 8: The results for additional Slovak LMs with Slovak Gender dataset. The rows are the same as in Table 1. | SS Gender | SS Gender Filter | CS Negation | CS Anti-stereotype
---|---|---|---|---
roberta-base
$cs\mu$ Original | $0.17\pm 0.081$ | $0.12\pm 0.11$ | $0.32\pm 0.33$ | $0.26\pm 0.38$
$cs\mu$ Control | $-0.049\pm 0.091$ | $-0.11\pm 0.11$ | $0.28\pm 0.37$ | $0.034\pm 0.4$
$csk\mu$ Original | $0.086\pm 0.046$ | $0.08\pm 0.056$ | - | -
$csk\mu$ Control | $-0.099\pm 0.052$ | $-0.14\pm 0.055$ | - | -
$cs+$ Original | $0.61\pm 0.06$ | $0.56\pm 0.079$ | $0.61\pm 0.11$ | $0.58\pm 0.12$
$cs+$ Control | $0.44\pm 0.061$ | $0.42\pm 0.079$ | $0.63\pm 0.11$ | $0.48\pm 0.12$
$cs\ \rho$ | $0.52$ | $0.58$ | $0.87$ | $0.76$
$csk\ \rho$ | $0.14$ | $0.13$ | - | -
$cs{-}csk\ \rho$ | $0.48$ | $0.49$ | - | -
bert-base-uncased
$cs\mu$ Original | $0.14\pm 0.085$ | $0.16\pm 0.1$ | $0.49\pm 0.32$ | $0.7\pm 0.35$
$cs\mu$ Control | $0.044\pm 0.086$ | $-0.056\pm 0.098$ | $0.63\pm 0.33$ | $0.42\pm 0.38$
$csk\mu$ Original | $0.093\pm 0.036$ | $0.1\pm 0.043$ | - | -
$csk\mu$ Control | $-0.044\pm 0.043$ | $-0.11\pm 0.054$ | - | -
$cs+$ Original | $0.59\pm 0.06$ | $0.61\pm 0.078$ | $0.57\pm 0.12$ | $0.61\pm 0.12$
$cs+$ Control | $0.51\pm 0.061$ | $0.45\pm 0.079$ | $0.59\pm 0.12$ | $0.54\pm 0.12$
$cs\ \rho$ | $0.54$ | $0.47$ | $0.88$ | $0.8$
$csk\ \rho$ | $0.11$ | $-0.042$ | - | -
$cs{-}csk\ \rho$ | $0.5$ | $0.45$ | - | -
distilbert-base-uncased
$cs\mu$ Original | $0.19\pm 0.085$ | $0.11\pm 0.11$ | $0.45\pm 0.36$ | $0.45\pm 0.4$
$cs\mu$ Control | $0.017\pm 0.086$ | $-0.13\pm 0.096$ | $0.51\pm 0.39$ | $0.39\pm 0.49$
$csk\mu$ Original | $0.12\pm 0.04$ | $0.11\pm 0.047$ | - | -
$csk\mu$ Control | $-0.047\pm 0.043$ | $-0.12\pm 0.057$ | - | -
$cs+$ Original | $0.59\pm 0.06$ | $0.57\pm 0.079$ | $0.61\pm 0.11$ | $0.61\pm 0.12$
$cs+$ Control | $0.49\pm 0.061$ | $0.41\pm 0.079$ | $0.59\pm 0.12$ | $0.57\pm 0.12$
$cs\ \rho$ | $0.62$ | $0.58$ | $0.97$ | $0.86$
$csk\ \rho$ | $0.092$ | $0.077$ | - | -
$cs{-}csk\ \rho$ | $0.53$ | $0.55$ | - | -
xlm-roberta-base
$cs\mu$ Original | $0.0061\pm 0.088$ | $-0.033\pm 0.11$ | $-0.28\pm 0.72$ | $-0.052\pm 0.32$
$cs\mu$ Control | $-0.11\pm 0.081$ | $-0.11\pm 0.11$ | $-0.058\pm 0.62$ | $-0.038\pm 0.33$
$csk\mu$ Original | $0.081\pm 0.04$ | $0.041\pm 0.038$ | - | -
$csk\mu$ Control | $-0.027\pm 0.038$ | $-0.083\pm 0.053$ | - | -
$cs+$ Original | $0.5\pm 0.061$ | $0.48\pm 0.08$ | $0.54\pm 0.12$ | $0.54\pm 0.12$
$cs+$ Control | $0.43\pm 0.061$ | $0.45\pm 0.079$ | $0.49\pm 0.12$ | $0.48\pm 0.12$
$cs\ \rho$ | $0.51$ | $0.58$ | $0.88$ | $0.84$
$csk\ \rho$ | $0.16$ | $-0.0075$ | - | -
$cs{-}csk\ \rho$ | $0.37$ | $0.18$ | - | -
albert-base-v2
$cs\mu$ Original | $0.1\pm 0.092$ | $0.15\pm 0.13$ | $-0.04\pm 0.58$ | $0.0021\pm 0.51$
$cs\mu$ Control | $-0.016\pm 0.1$ | $-0.019\pm 0.11$ | $0.0098\pm 0.54$ | $-0.014\pm 0.5$
$csk\mu$ Original | $0.069\pm 0.028$ | $0.057\pm 0.034$ | - | -
$csk\mu$ Control | $-0.039\pm 0.027$ | $-0.069\pm 0.031$ | - | -
$cs+$ Original | $0.56\pm 0.061$ | $0.62\pm 0.077$ | $0.56\pm 0.12$ | $0.55\pm 0.12$
$cs+$ Control | $0.42\pm 0.061$ | $0.44\pm 0.079$ | $0.59\pm 0.12$ | $0.51\pm 0.12$
$cs\ \rho$ | $0.59$ | $0.56$ | $0.96$ | $0.86$
$csk\ \rho$ | $0.14$ | $0.045$ | - | -
$cs{-}csk\ \rho$ | $0.34$ | $0.17$ | - | -
bert-base-multilingual-cased
$cs\mu$ Original | $0.093\pm 0.083$ | $0.098\pm 0.12$ | $0.099\pm 0.35$ | $0.2\pm 0.23$
$cs\mu$ Control | $0.015\pm 0.08$ | $0.042\pm 0.11$ | $0.33\pm 0.27$ | $0.27\pm 0.29$
$csk\mu$ Original | $0.042\pm 0.032$ | $0.044\pm 0.035$ | - | -
$csk\mu$ Control | $-0.0055\pm 0.035$ | $0.0057\pm 0.041$ | - | -
$cs+$ Original | $0.55\pm 0.061$ | $0.58\pm 0.079$ | $0.5\pm 0.12$ | $0.67\pm 0.11$
$cs+$ Control | $0.47\pm 0.061$ | $0.48\pm 0.08$ | $0.6\pm 0.11$ | $0.55\pm 0.12$
$cs\ \rho$ | $0.74$ | $0.74$ | $0.86$ | $0.76$
$csk\ \rho$ | $0.23$ | $0.17$ | - | -
$cs{-}csk\ \rho$ | $0.28$ | $0.24$ | - | -
Table 9: The results for additional English LMs. The rows are the same as in Table 2. | gerulata/slovakbert | xlm-roberta-base | bert-base-multilingual-cased
---|---|---|---
$cs\mu$ Original | $0.074\pm 0.11$ | $0.097\pm 0.22$ | $0.086\pm 0.18$
$cs\mu$ Control | $0.086\pm 0.11$ | $0.053\pm 0.22$ | $-0.17\pm 0.18$
$csk\mu$ Original | $0.08\pm 0.057$ | $0.089\pm 0.083$ | $-0.017\pm 0.072$
$csk\mu$ Control | $0.045\pm 0.067$ | $0.051\pm 0.081$ | $-0.087\pm 0.055$
$cs+$ Original | $0.51\pm 0.083$ | $0.59\pm 0.082$ | $0.5\pm 0.083$
$cs+$ Control | $0.59\pm 0.082$ | $0.47\pm 0.083$ | $0.45\pm 0.083$
$cs\ \rho$ | $0.77$ | $0.66$ | $0.63$
$csk\ \rho$ | $0.79$ | $0.63$ | $0.43$
$cs{-}csk\ \rho$ | $0.19$ | $0.33$ | $0.36$
Table 10: The results for additional Slovak LMs with Slovak Gender dataset. The rows are the same as in Table 2. | SS Gender | SS Gender Filter | SS Race | SS Profession
---|---|---|---|---
roberta-base
$f\mu$ | $0.18\pm 0.065$ | $0.22\pm 0.078$ | $0.095\pm 0.013$ | $0.25\pm 0.017$
$f+$ | $0.6\pm 0.06$ | $0.64\pm 0.076$ | $0.54\pm 0.01$ | $0.62\pm 0.011$
$f{-}ss\ \rho$ | $0.2$ | $0.12$ | $0.082$ | $0.32$
$f{-}ss$ agreement | $0.56$ | $0.56$ | $0.54$ | $0.62$
$f{-}cs\ \rho$ | $0.27$ | $0.27$ | $0.24$ | $0.24$
$f{-}cs$ agreement | $0.59$ | $0.59$ | $0.58$ | $0.64$
bert-base-uncased
$f\mu$ | $0.14\pm 0.054$ | $0.21\pm 0.073$ | $0.11\pm 0.0098$ | $0.19\pm 0.016$
$f+$ | $0.62\pm 0.059$ | $0.68\pm 0.074$ | $0.57\pm 0.0099$ | $0.6\pm 0.011$
$f{-}ss\ \rho$ | $0.13$ | $0.12$ | $0.18$ | $0.34$
$f{-}ss$ agreement | $0.54$ | $0.52$ | $0.57$ | $0.62$
$f{-}cs\ \rho$ | $0.37$ | $0.44$ | $0.3$ | $0.25$
$f{-}cs$ agreement | $0.61$ | $0.6$ | $0.61$ | $0.62$
distilbert-base-uncased
$f\mu$ | $0.16\pm 0.053$ | $0.23\pm 0.07$ | $0.11\pm 0.0093$ | $0.2\pm 0.014$
$f+$ | $0.63\pm 0.059$ | $0.69\pm 0.074$ | $0.58\pm 0.0099$ | $0.62\pm 0.011$
$f{-}ss\ \rho$ | $0.18$ | $0.1$ | $0.19$ | $0.33$
$f{-}ss$ agreement | $0.6$ | $0.55$ | $0.57$ | $0.64$
$f{-}cs\ \rho$ | $0.37$ | $0.41$ | $0.33$ | $0.3$
$f{-}cs$ agreement | $0.61$ | $0.62$ | $0.62$ | $0.64$
xlm-roberta-base
$f\mu$ | $0.11\pm 0.05$ | $0.12\pm 0.067$ | $0.069\pm 0.01$ | $0.14\pm 0.013$
$f+$ | $0.63\pm 0.059$ | $0.67\pm 0.075$ | $0.54\pm 0.01$ | $0.59\pm 0.011$
$f{-}ss\ \rho$ | $0.036$ | $0.031$ | $0.14$ | $0.32$
$f{-}ss$ agreement | $0.57$ | $0.59$ | $0.55$ | $0.62$
$f{-}cs\ \rho$ | $0.21$ | $0.11$ | $0.2$ | $0.18$
$f{-}cs$ agreement | $0.56$ | $0.54$ | $0.57$ | $0.6$
albert-base-v2
$f\mu$ | $0.11\pm 0.036$ | $0.13\pm 0.045$ | $0.072\pm 0.011$ | $0.19\pm 0.015$
$f+$ | $0.64\pm 0.059$ | $0.66\pm 0.075$ | $0.56\pm 0.0099$ | $0.61\pm 0.011$
$f{-}ss\ \rho$ | $0.072$ | $0.0067$ | $0.15$ | $0.29$
$f{-}ss$ agreement | $0.56$ | $0.53$ | $0.56$ | $0.61$
$f{-}cs\ \rho$ | $0.17$ | $0.14$ | $0.21$ | $0.17$
$f{-}cs$ agreement | $0.61$ | $0.61$ | $0.62$ | $0.62$
bert-base-multilingual-cased
$f\mu$ | $0.047\pm 0.043$ | $0.038\pm 0.049$ | $0.033\pm 0.011$ | $0.15\pm 0.013$
$f+$ | $0.53\pm 0.061$ | $0.51\pm 0.08$ | $0.53\pm 0.01$ | $0.6\pm 0.011$
$f{-}ss\ \rho$ | $0.15$ | $0.077$ | $0.14$ | $0.32$
$f{-}ss$ agreement | $0.57$ | $0.53$ | $0.54$ | $0.6$
$f{-}cs\ \rho$ | $0.19$ | $0.15$ | $0.16$ | $0.2$
$f{-}cs$ agreement | $0.55$ | $0.54$ | $0.56$ | $0.59$
Table 11: The results for additional English LMs. The rows are the same as in Table 3. | gerulata/slovakbert | xlm-roberta-base | bert-base-multilingual-cased
---|---|---|---
$f\mu$ | $0.035\pm 0.04$ | $0.038\pm 0.071$ | $0.069\pm 0.066$
$f+$ | $0.54\pm 0.083$ | $0.53\pm 0.083$ | $0.58\pm 0.082$
$f{-}ss\ \rho$ | $0.022$ | $0.18$ | $0.26$
$f{-}ss$ agreement | $0.51$ | $0.57$ | $0.53$
$f{-}cs\ \rho$ | $0.024$ | $-0.085$ | $0.21$
$f{-}cs$ agreement | $0.53$ | $0.52$ | $0.53$
Table 12: The results for additional Slovak LMs with Slovak Gender dataset.
The rows are the same as in Table 3.


# Probabilistic work extraction on a classical oscillator beyond the second
law

Nicolas Barros  0009-0000-1348-7725 Univ Lyon, ENS de Lyon, CNRS, Laboratoire
de Physique, F-69342 Lyon, France  Sergio Ciliberto  0000-0002-4366-6094 Univ
Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France
Ludovic Bellon  0000-0002-2499-8106<EMAIL_ADDRESS>Univ Lyon, ENS
de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France

###### Abstract

We demonstrate experimentally that, applying optimal protocols which drive the
system between two equilibrium states characterized by a free energy
difference $\Delta F$, we can maximize the probability of performing the
transition between the two states with a work $W$ smaller than $\Delta F$. The
second law holds only on average, resulting in the inequality $\langle
W\rangle\geq\Delta F$. The experiment is performed using an underdamped
oscillator evolving in a double-well potential. We show that with a suitable
choice of parameters the probability of obtaining trajectories with
$W\leq\Delta F$ can be larger than $90\,\%$. Very fast protocols are a key
feature to obtain these results which are explained in terms of the Jarzynski
equality.

##  I Introduction

Numerous experimental platforms that act on the micro and nano scales allow us
to explore the laws of thermodynamics for systems with few degrees of freedom
coupled with thermal baths [1]. Typical examples are experiments in colloids
[2, 3, 4, 5, 6, 7], electric circuits [8], single electron transistors [9,
10], mechanical devices [11, 12] and single molecules [13, 14]. In such
systems, fluctuations of physical quantities play a fundamental role, contrary
to classical thermodynamics which mostly considers averaged quantities.
Stochastic thermodynamics provides the suited framework to describe single
realizations of thermodynamic transformation, and the associated work and heat
distribution. In particular, the second law does not apply at the level of a
single realization, and one can observe local “violations” due to the
stochastic nature of the system, where the system can extract work, or gain
free energy, at no cost to the operator and with no information feedback
(illustration in Fig. 1) [15, 16, 17, 18].

In a system driven by an external control parameter $\lambda$ from an initial
to a final equilibrium state resulting in a free energy difference $\Delta F$
between the two states, the probability distribution of the work performed
when changing $\lambda$ is constrained by Jarzynski’s equality [19]

$\langle\mathrm{e}^{-\beta W}\rangle=\mathrm{e}^{-\beta\Delta F},$ (1)

where $\beta=1/k_{B}T$, with $T$ the temperature of the system and $k_{B}$ the
Boltzmann constant. The expression of the second law can be recovered from the
convexity of the exponential function: $\langle W\rangle\geq\Delta F$. We are
interested here in realizations where the free energy of the system can be
increased spending an amount of work $W<\Delta F$. The free energy stored in
the system can be later used, resulting in probabilistic work extraction from
the device.

Figure 1: Probability distribution function of the work $P(W)$ during a
transformation. (a) In classical macroscopic thermodynamics, fluctuations are
usually negligible compared with the mean value. (b) However, in stochastic
thermodynamics, such fluctuations can lead to local “violations” of the second
law: several realisations of a transformation can be performed with a work $W$
smaller than the free energy difference $\Delta F$ (hatched area), even if in
average $\langle W\rangle\geq\Delta F$. Note that in this sketch, the
horizontal axis is rescaled between (a) and (b).

These realizations have been theoretically [10] and experimentally studied [9]
with a single electron transistor with discrete energy levels, obtaining a
probability of extracting work up to $65\text{\,}\%$. Using a mechanical
oscillator, we want to illustrate this behavior using a fully classical
continuous system. Our goal is to experimentally obtain an arbitrary large
probability of having realizations with $W<\Delta F$. Also, we approach the
optimal work distribution predicted by [10] to observe such events at a fixed
mean cost of energy.

##  II Experimental setup and protocol

The experimental setup is a microcantilever, which behaves in the absence of
external forces as an underdamped harmonic oscillator of stiffness $k$,
resonance frequency $f_{0}=1200Hz$ and quality factor $Q=10$. The deflection
$x$ of the cantilever is measured by interferometry [20]. The oscillator is in
equilibrium with the surrounding air at room temperature $T$ and subject to
thermal fluctuations. The variance of $x$ is $\sigma^{2}=\langle
x^{2}\rangle=k_{B}T/k\sim$1\text{\,}\mathrm{n}\mathrm{m}^{2}$$. $\sigma$ is
used as the unit length, and from now on all positions are expressed as
dimensionless quantities $z=x/\sigma$, and energies in units of $k_{B}T$
(hence taking $\beta=1$ in Jarzynski’s equality, Eq. 1).

Figure 2: Sketch of the experimental setup. We record the deflection $z$ of a
conductive cantilever measured with high precision by a differential
interferometer [20]. Depending on the comparison of $z$ to a tunable threshold
$\lambda$, a constant voltage $V=\pm V_{0}$ is applied by a fast feedback
loop, creating an electrostatic force $F_{\mathrm{el}}$ on the cantilever
towards a nearby electrode at voltage $V_{1}\gg V_{0}$.

To tune the potential experienced by the resonator, we use an electrostatic
force acting on the cantilever. A time-dependent virtual potential can be
created by a fast feedback loop [11, 21] that adjusts the voltage $V$ applied
to the cantilever (thus the electrostatic force) depending on the measured
position $z$ (see Fig. 2). We implement the following simple rule: $V=-V_{0}$
if the position is below a threshold $\lambda$, and $V=+V_{0}$ otherwise. This
creates an asymmetric double well, illustrated in Fig. 3. Two parameters are
available to tune the potential: $\lambda$ sets the barrier position and
$V_{0}$ sets the centers of the two wells in $\pm z_{0}\propto\pm V_{0}$.

Theoretically, the potential energy constructed by this feedback is:

$\begin{split}U(z,\lambda,z_{0})=&\frac{1}{2}\big{(}z-S(z-\lambda)z_{0}\big{)}^{2}\\\
&+\lambda z_{0}\big{(}S(z-\lambda)+S(\lambda)\big{)},\end{split}$ (2)

where $S$ is the sign function: $S(z-\lambda)=-1$ if $z<\lambda$ and
$S(z-\lambda)=1$ if $z>\lambda$. To illustrate the validity of this model for
$U$, we record a long time trace of the position in a static potential to
evaluate the probability distribution function (pdf) of the position $P(z)$.
We then reconstruct the double well using Boltzmann’s prescription at
equilibrium:

$\displaystyle P(z)$
$\displaystyle=\frac{1}{Z(\lambda,z_{0})}\mathrm{e}^{-U(z,\lambda,z_{0})},$
(3a) $\displaystyle Z(\lambda,z_{0})$
$\displaystyle=\int_{-\infty}^{+\infty}\mathrm{e}^{-U(z,\lambda,z_{0})}\mathrm{d}z,$
(3b)

with $Z$ the partition function. As plotted in Fig. 3, the model is an
excellent description of the effective potential evaluated through
$U(z)=-\ln[P(z)/P(-S(\lambda)z_{0})]$.

Figure 3: Double well potentials. The potential energy of the cantilever is
the juxtaposition of two harmonic wells centered in $\pm z_{0}=\pm 1.91$, the
switch between the two occurring when $z=\lambda$. The two examples plotted
here correspond to $\lambda_{i}=0.44$ and $\lambda_{f}=0.86$. The measured
potential $U(z)$ data is inferred from Eq. 3 and the pdf of the experimental
positions $z$ during a long acquisition. The black dashed lines are best fit
to Eq. 2, leading to the aforementioned values of $z_{0}$ and $\lambda$. The
transformation we apply corresponds to a step of $\lambda$ from $\lambda_{i}$
to $\lambda_{f}$, leaving the lower well untouched while raising the upper one
by $\Delta U=2(\lambda_{f}-\lambda_{i})x_{0}$.

To observe local “violations” of the second law, we design the following
protocol between an initial state (all quantities labeled by the subscript i)
and a final state (subscript f). First, the cantilever evolves at equilibrium
in an initial double-well potential $U_{i}(z)=U(z,\lambda_{i},z_{0})$. Then,
we instantaneously increase the threshold $\lambda$ between the two wells,
from $\lambda_{i}$ to $\lambda_{f}$. Finally, the cantilever is left in the
final potential $U_{f}(z)=U(z,\lambda_{f},z_{0})$. The well centers are left
unchanged during the protocol, $z_{0}$ keeping the same value. An example of
the time trace of one realisation of such protocol is plotted in Fig. 4.

Figure 4: Trajectory of an underdamped oscillator evolving in a time-dependent
asymmetric double-well potential. The position $z$ is expressed in units of
standard deviation $\sigma$ in a single well. The center of the wells is kept
constant at $\pm z_{0}=\pm 1.91$, but the commutation threshold $\lambda$ is
changed from $\lambda_{i}=0.44$ to $\lambda_{f}=0.86$ at
$t=$150\text{\,}\mathrm{m}\mathrm{s}$$.

Initial and final potentials are represented in Fig. 3. Each potential is
measured from the equilibrium pdf $P(z)$ in the initial and final state, then
fitted with very good accuracy using Eq. 2. The experimental values of the
parameters $\lambda_{i}$, $\lambda_{f}$ and $z_{0}$ are deduced from the fit.
Potential wise, the protocol amounts to leaving the lower well unaffected
while raising the upper one. The work $W$ performed will be $0$ most of the
time (anytime the system is in the lower well, which is likely), while the
free energy difference is positive (since $U(x)$ is globally increasing): the
probability of observing $W<\Delta F$ should thus be high.

The values of $z_{0}=1.91$ and $\lambda_{f}=0.86$ are the same for all
experiments. We tune the initial threshold $\lambda_{i}$ with values going
from 0 to 0.75. For each one of them, the protocol detailed in Fig. 4 is
repeated around $N\sim 1000$ times. This allows us to obtain a good enough
statistics to estimate the pdf $P(z)$, and further on the work distribution.

##  III Analysis

Following the classical convention of stochastic thermodynamics [16], we
define the work $W$ done by the operator through the variations of the
external parameters tuning the potential $U(z,\lambda,z_{0})$:

$W=\int\frac{\partial U}{\partial\lambda}\dot{\lambda}\mathrm{d}t.$ (4)

Since $z_{0}$ is kept constant in all protocols, we do not consider the term
in $\dot{z}_{0}$ which is always zero here. $W$ is computed using the recorded
trajectories $z(t)$ and Eq. 2. Since the variation of $\lambda(t)$ is a step
function, the work can be equivalently computed as $W=U_{f}(z)-U_{i}(z)$ using
the value of $z$ at the moment of the switch. From this expression we can also
infer the theoretical expectation for $\langle W\rangle$:

$\displaystyle\langle W\rangle$
$\displaystyle=\int_{-\infty}^{+\infty}\big{[}U_{f}(z)-U_{i}(z)\big{]}\frac{1}{Z_{i}}\mathrm{e}^{-U_{i}(z)}\mathrm{d}z,$
(5a)
$\displaystyle=\frac{2z_{0}\mathrm{e}^{-2\lambda_{i}z_{0}}}{Z_{i}}\big{[}f(\lambda_{f}-z_{0})-f(\lambda_{i}-z_{0})\big{]},$
(5b)

where $f$ is defined by

$f(u)=\sqrt{\pi/2}\,u\,\mathrm{erfc}\left(-u/\sqrt{2}\right)-\mathrm{e}^{-u^{2}/2}.$
(6)

$Z_{i}=Z(\lambda_{i},z_{0})$ is the partition function in the initial state,
which can be computed from Eq. 3b as:

$\begin{split}Z(\lambda,z_{0})=\sqrt{\frac{\pi}{2}}\bigg{[}2&-\mathrm{erfc}\bigg{(}\frac{\lambda+z_{0}}{\sqrt{2}}\bigg{)}\\\
&+\mathrm{e}^{-2\lambda z_{0}}\mathrm{erfc}\bigg{(}\frac{\lambda-
z_{0}}{\sqrt{2}}\bigg{)}\bigg{]}.\end{split}$ (7)

The free energy difference in the system $\Delta F=F_{f}-F_{i}$ can be
computed in two different ways. For a given protocol, the work distribution
obeys Jarzynski’s equality (Eq. 1), thus $\Delta F=-\ln\langle e^{-W}\rangle$
can be deduced from the experimental work distribution with good enough
statistics. The second approach is to use the partition function $Z$ at
equilibrium: since the free energy of the system is $F=-\ln(Z)$, the free
energy difference $\Delta F=\ln(Z_{i}/Z_{f})$ can be theoretically directly
calculated from the parameters of the initial and final potentials.

Figure 5:  (a) Work distribution for $N=968$ protocols, corresponding to the
protocol in Fig. 3 ($\lambda_{i}=0.44$, $\lambda_{f}=0.86$). We also report
the mean work $\langle W\rangle=0.23\pm 0.02$ and free energy difference
$\Delta F=0.14\pm 0.01$ computed using Jarzynski’s equality. We observe a
transient violation of the second law in more than 80% of the protocols (note
the vertical log scale). (b) Mean work $\langle W\rangle$ ($\circ$, top curve)
and free energy difference $\Delta F$ ($+$, bottom curve) and for all initial
conditions $\lambda_{i}$. Experiment (markers with error bars corresponding to
one standard deviation of the statistical uncertainties) and theory (dashed
lines) are in good agreement. The second law of thermodynamics $\langle
W\rangle\geq\Delta F$ always holds, though the probability of observing the
contrary on the single trajectory reaches over $90\text{\,}\%$ for the highest
values of $\lambda_{i}$ ($\circ$ marker, experiment, and dashed blue line,
theory, right vertical scale). The probability of observing a zero work
$P(W=0)$ ($\times$) almost coincide with the one of transiently violating the
second law, and is upper bounded by $\mathrm{e}^{-\Delta F}$ (dotted blue
line, right vertical scale), as predicted by Eq. 10 [22].

##  IV Results

The protocol is repeated $N\sim 1000$ times for 10 values of $\lambda_{i}$
ranging from 0 to 0.75. For each $\lambda_{i}$, we compute the work
distribution. An example for $\lambda_{i}=0.44$ is plotted in Fig. 5(a), where
we also report the mean work $\langle W\rangle$ performed by the driving, and
the difference in free energy $\Delta F$. The work distribution consists
mostly of two narrow peaks. The first one, for $W=0$, corresponds to the cases
where the cantilever was in the lower well ($z<\lambda_{i}$) of the potential
during the step of $\lambda$. Since there is no change in the lower part of
the potential, there is no energy cost. The second peak comes from the cases
where $z>\lambda_{f}$: in this area, the potential is shifted by $\Delta
U=2z_{0}(\lambda_{f}-\lambda_{i})$, and driving the parameter $\lambda$
implies this amount of work. We also observe some intermediate values,
corresponding to $\lambda_{i}<z<\lambda_{f}$: the cantilever is initially in
the upper well but ends in the lower one due to the step in in $\lambda$,
resulting in an intermediate energy change $\Delta U=2z_{0}(z-\lambda_{i})$,
where $z$ is the value of the deflection at the time of the switch.

As illustrated in Fig. 5(b), the second law is always satisfied ($\langle
W\rangle>\Delta F$), but we manage to observe an almost arbitrary large
proportion of transient violations ($W<\Delta F$). Indeed, by tuning the
initial asymmetry of the potential, we can increase the probability of being
in the lower well ($z<\lambda_{i}$) during the switch. The probability of
observing a transient violation of the second law $P(W<\Delta F)$ can thus be
arbitrary large. With our choice of parameters, for values of $\lambda_{i}$
very close to the final value of $\lambda_{f}$, we manage to measure values of
$P(W<\Delta F)$ greater than 90% . Our experimental results are very close to
the theoretical expectation:

$\displaystyle\\!\\!P(W<\Delta F)$ $\displaystyle=\int_{0}^{\Delta
F}P(W)\mathrm{d}W=\int_{-\infty}^{\lambda_{i}+\Delta
F/(2z_{0})}\frac{1}{Z_{i}}\mathrm{e}^{-U_{i}(z)}\mathrm{d}z$ (8)
$\displaystyle=\frac{1}{Z(\lambda_{i},z_{0})}\sqrt{\frac{\pi}{2}}\bigg{[}2-\mathrm{erfc}\bigg{(}\frac{\lambda_{i}+z_{0}}{\sqrt{2}}\bigg{)}+\mathrm{e}^{-2\lambda_{i}z_{0}}\mathrm{erfc}\bigg{(}\frac{\lambda_{i}-z_{0}}{\sqrt{2}}\bigg{)}-\mathrm{e}^{-2\lambda_{i}z_{0}}\mathrm{erfc}\bigg{(}\frac{\Delta
F/(2z_{0})+\lambda_{i}-z_{0}}{\sqrt{2}}\bigg{)}\bigg{]}.$ (9)

However, the free energy $\Delta F$ decreases when increasing $\lambda_{i}$:
work extraction is more and more likely, but the gain with respect to $\Delta
F$ decreases.

Another theoretical result that we can probe with our experiment is the
inequality [22]:

$P(W\leq 0)\leq\mathrm{e}^{-\Delta F}.$ (10)

In our case, $P(W\leq 0)=P(W=0)$ since $W$ cannot be negative. Moreover, this
probability is very close to $P(W\leq\Delta F)$ since the work distribution
consists mainly of two peaks: one in $0$ and the other above $\Delta F$. In
the experiments, the two probabilities coincide within statistical
uncertainties, as shown in Fig. 5(b). In this same figure, we plot the upper
bound given by Eq. 10, which is indeed confirmed in our experiment, and close
to be saturated.

## Conclusion

We have shown, using a fully classical continuous mechanical system, how we
can observe an arbitrary large number of apparent “violations” of the second
law of thermodynamics, while being consistent with the rules of stochastic
thermodynamics. We show a clear trade-off between this probability and the
free energy gained during those events. We almost reach the theoretical limit
of $95\%$ probability (for our set of parameters) of having anomalous
trajectories with $W<\Delta F$. This result is made possible by the specific
way in which the potential is changed during the protocol. Indeed the
stiffness of the two wells does not change and only the minimum of the upper
well is raised. This protocol produces a work probability distribution with
mainly two Dirac functions that matches the optimal distribution described in
Ref. [10] to maximize the work extraction probability. One Dirac peak is
centered in zero and corresponds to the trajectories that start in the lower
well. The other is centered to a positive value of the work and corresponds to
the trajectories starting on the upper well. In our experiment the relative
amplitude of the two peaks, which determines the amount of “anomalous”
trajectories, can be tuned by changing the minimum positions of the wells
through the value of $\lambda$. In this way we have transformed for the
transition properties a continuous classical system to a two levels system,
showing similarities with the one of ref. [9]. In this context, the fast
switch between the initial and final state is a key ingredient of the
protocol, again as proposed in Ref. [10]. Indeed a slow ramp would broaden the
peaks of the work distribution, resulting in a situation similar to Fig. 1(b).
For such symmetric distributions, where the mean and the median are equal,
observing $W<\Delta F$ is unlikely: the probability is smaller than
$50\text{\,}\%$, this limit being reached in the reversible limit. Let us to
conclude by saying that in spite of the fact there is an energy gain for
$90\%$ of the trajectories, the total mean remains positive and in order to
use this energy surplus one should introduce a demon which selects the good
trajectories, i.e. those starting on the lower well.

## Acknowledgements

This work has been supported by project ANR-22-CE42-0022. We thank C.
Jarzynski for enlightening physical discussions.

The data supporting this study are openly available in Ref. 23.

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