Abstract
As artificial intelligence becomes more embedded into daily activities, it is imperative to ensure models perform well for all subgroups. This is particularly important when models include underprivileged populations. Binary fairness metrics, which compare model performance for protected groups to the rest of the model population, are an important way to guard against unwanted bias. However, a significant drawback of these binary fairness metrics is that they require protected group membership attributes. In many practical scenarios, protected status for individuals is sparse, unavailable, or even illegal to collect. This paper extends binary fairness metrics from deterministic membership attributes to their surrogate counterpart under the probabilistic setting. We show that it is possible to conduct binary fairness evaluation when exact protected attributes are not immediately available but their surrogate as likelihoods is accessible. Our inferred metrics calculated from surrogates are proved to be valid under standard statistical assumptions. Moreover, we do not require the surrogate variable to be strongly related to protected class membership; inferred metrics remain valid even when membership in the protected and unprotected groups is equally likely for many levels of the surrogate variable. Finally, we demonstrate the effectiveness of our approach using publicly available data from the Home Mortgage Disclosure Act and simulated benchmarks that mimic real-world conditions under different levels of model disparity.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aitken, A.C.: On least squares and linear combination of observations. Proc. R. Soc. Edinb. 55, 42–48 (1936)
Andrus, M., Spitzer, E., Brown, J., Xiang, A.: What we can’t measure, we can’t understand: challenges to demographic data procurement in the pursuit of fairness. In: Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency, pp. 249–260 (2021)
Bellamy, R.K.E., et al.: AI Fairness 360: an extensible toolkit for detecting, understanding, and mitigating unwanted algorithmic bias (2018). https://arxiv.org/abs/1810.01943
Box, G.E.: Use and abuse of regression. Technometrics 8(4), 625–629 (1966)
Caton, S., Haas, C.: Fairness in machine learning: a survey. arXiv preprint arXiv:2010.04053 (2020)
Chen, J., Kallus, N., Mao, X., Svacha, G., Udell, M.: Fairness under unawareness: assessing disparity when protected class is unobserved. In: Proceedings of the Conference on Fairness, Accountability, and Transparency, pp. 339–348 (2019)
Chenevert, R., Gottschalck, A., Klee, M., Zhang, X.: Where the wealth is: the geographic distribution of wealth in the united states. US Census Bureau (2017)
Department, U.F.R.: Federal fair lending regulations and statutes (2020). https://www.federalreserve.gov/boarddocs/supmanual/cch/fair_lend_over.pdf. Accessed 04 Sept 2020
Din, A., Wilson, R.: Crosswalking zip codes to census geographies. Cityscape 22(1), 293–314 (2020)
Duris, F., et al.: Mean and variance of ratios of proportions from categories of a multinomial distribution. J. Stat. Distrib. Appl. 5(1), 1–20 (2018)
Efron, B., Tibshirani, R.: Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Stat. Sci. 1, 54–75 (1986)
Elliott, M.N., Morrison, P.A., Fremont, A., McCaffrey, D.F., Pantoja, P., Lurie, N.: Using the census bureau’s surname list to improve estimates of race/ethnicity and associated disparities. Health Serv. Outcomes Res. Method. 9(2), 69–83 (2009)
Farrar, D.E., Glauber, R.R.: Multicollinearity in regression analysis: the problem revisited. In: The Review of Economic and Statistics, pp. 92–107 (1967)
Federal Financial Institutions Examination Council: Home mortgage disclosure act snapshot national loan level dataset. Technical report, U.S. Government (2018). https://ffiec.cfpb.gov/data-publication/snapshot-national-loan-level-dataset/2018
Gal, Y., Ghahramani, Z.: Dropout as a Bayesian approximation: representing model uncertainty in deep learning. In: International Conference on Machine Learning, pp. 1050–1059. PMLR (2016)
Hays, W.: The algebra of expectations. In: Statistics, p. 630. CBS College Publishing, Holt Rhinehart Winston New York (1981)
Heitjan, D.F.: Inference from grouped continuous data: a review. Stat. Sci. 4(2), 164–179 (1989)
Kallus, N., Mao, X., Zhou, A.: Assessing algorithmic fairness with unobserved protected class using data combination. Manage. Sci. 68(3), 1959–1981 (2022)
of Labor, D.: Uniform guidelines on employee selection procedures (1978). https://uniformguidelines.com/questionandanswers.html. Accessed 05 Sept 2020
Michalský, F., Kadioglu, S.: Surrogate ground truth generation to enhance binary fairness evaluation in uplift modeling. In: 20th IEEE International Conference on ML and Applications, ICMLA 2021, USA, 2021, pp. 1654–1659. IEEE (2021)
Papoulis, A.: Expected value; dispersion; moments (1984)
Racicot, T., Khoury, R., Pere, C.: Estimation of uncertainty bounds on disparate treatment when using proxies for the protected attribute. In: Canadian Conference on AI (2021)
U.S. Department of Housing and Urban Development: Fair housing rights and obligations (2020). https://www.hud.gov/program_offices/fair_housing_equal_opp/fair_housing_rights_and_obligations. Accessed 04 Sept 2020
U.S. Equal Employment Opportunity Commission: Prohibited employment policies/practices (2020). https://www.eeoc.gov/prohibited-employment-policiespractices. Accessed 04 Sept 2020
VanderWeele, T.J., Shpitser, I.: On the definition of a confounder. Ann. Stat. 41(1), 196 (2013)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendix - Comparison to Weighted Fairness Statistic
Appendix - Comparison to Weighted Fairness Statistic
Our inferred metrics are similar in approach to an estimator described in [6]. In this section, we re-write our inferred metrics and the weighted estimator so they can be compared directly and present a mathematical argument for why the weighted estimator is biased toward 0 under the regularity conditions described.
First, our estimator \(m(X^\top )-m(X^\bot )\) is derived from the WOLS estimator of the value \(\beta _1\) from Eq. 5.
where \(\bar{m}\) is the overall mean for the model metric and \(\bar{P}(x\in X^\top )\) is the overall mean for the probability of being in the protected group.
The weighted estimator described in [6] is:
where
-
m(x) is the value of the model metric for each individual (e.g. if the m is statistical parity, \(m(x)=I(ML(x)=1)\))
-
\(\sum _x\) indicates a sum over all N individuals for which we are calculating fairness metrics
-
\(P_z(x\in X^\top )\) is the probability that each individual is in the protected group given their surrogate class membership \(z \in Z\).
In the proof below, we re-write these equations to show that they are the same except for one term in the denominator. Specifically, we re-write our inferred metric as:
We re-write the weighted estimator from [6] as:
where \(N=\sum _z n_z\) (N is the total number of individuals for which we are calculating fairness metrics).
Equation 10 and Eq. 11 are the same except for the first term in the denominator. We argue here that this difference implies that the weighted estimator is biased toward 0 under the conditions described in (Sect. 3). This means that the weighted estimator will show smaller differences between groups than are actually present in the data.
First, note that \(P_z(x\in X^\top )\) is a probability, and therefore bounded between (0,1)
This means that the sign of the weighted estimator (whether it is negative or positive) is determined by the numerator of the equation.
Now, because \(P_z(x\in X^\top )\) is a probability,
This shows that the first term in the denominator is smaller for our inferred estimator, and therefore:
which means:
In (Sect. 3) we refer to a set of conditions where WOLS is unbiased that follows from [1] and the Gauss-Markov theorem. The weighted estimator is always smaller in absolute value and must therefore be biased toward 0 under the same conditions.
1.1 Re-Writing the Weighted Estimator
In order to compare the weighted estimator with our inferred estimator, we re-write the weighted estimator for the case where there are two groups, and one surrogate variable Z that acts as a predictor. Now, \(P_z(x\in X^\bot )=1-P_z(x \in X^\top )\), so that:
Multiply each of these fractions to get a common denominator.
Then, starting with the numerator, we expand the parentheses and distribute the sums, which gives the following:
The second and fourth terms cancel, so that:
Following the same process for the denominator gives us the following form for the weighted estimator:
1.2 Re-Writing the Inferred Estimator
We can follow the same process to re-write the estimator for the difference between \(m_{wols}(X^\top )-m_{wols}(X^\bot )\), and express our inferred fairness metric in terms of the individual values m(x).
As before, start with the numerator, expand the terms in parentheses and distribute the sums, which gives us the following expression.
Observe the following:
-
We require m to be an arithmetic mean, therefore, \(m_z=\frac{1}{n_z}\sum _z m(x)\), and \(n_z m_z=\sum _{z}m_{x}\)
-
\(\bar{m}=\frac{1}{N} \sum _x m(x)\)
-
\(\bar{P}(x\in X^\top )=\frac{1}{N}\sum _x P(x\in X^\top )=\frac{1}{N}\sum _z n_z P_z(x\in X^\top )\)
Taking each term in the numerator separately, we re-write them as:
-
\(\sum _z n_z m_z\bar{P}(x\in X^\top )=\sum _z n_z(m_z P_z(x\in X^\top ))= \sum _x m(x) P_z(x\in X^\top )\)
-
\(\bar{m}\sum _z n_z P_z(x\in X^\top )=\frac{1}{N}\sum _x m(x) \sum _x P(x\in X^\top )= N\bar{m}\bar{P}(x\in X^\top )\)
-
\(\bar{P}\sum _z n_z m_z=\frac{1}{N}\sum _x P(x\in X^\top ) \sum _x m(x) = N\bar{m}\bar{P}(x\in X^\top )\)
-
\(\bar{m}\bar{P}(x\in X^\top )\sum _z n_z=N\bar{m}\bar{P}(x\in X^\top )\)
This lets us collect three of the four terms in the numerator and leaves us with:
For the denominator, we again expand the parentheses and collect the sums to give the following:
Again, taking each term separately, we simplify as follows:
-
\(\sum _z n_z P_z(x\in X^\top )^2 = \sum _z \sum _{x\in z} P_z(x\in X^\top )^2= \sum _x P_z(x\in X^\top )^2\)
-
\(- 2\sum _z n_z P_z(x\in X^\top )\bar{P}(x\in X^\top )=-2N\bar{P}(x\in X^\top )^2\)
-
\(\sum _z n_z \bar{P}(x\in X^\top )^2=\bar{P}_(x\in X^\top )^2\sum _z n_z = N\bar{P}(x\in X^\top )^2\)
This gives us the expression:
Multiplying the above fraction by \(\frac{N}{N}\), gives us the form of the equation as written in (10).
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Thielbar, M., Kadıoğlu, S., Zhang, C., Pack, R., Dannull, L. (2023). Surrogate Membership for Inferred Metrics in Fairness Evaluation. In: Sellmann, M., Tierney, K. (eds) Learning and Intelligent Optimization. LION 2023. Lecture Notes in Computer Science, vol 14286. Springer, Cham. https://doi.org/10.1007/978-3-031-44505-7_29
Download citation
DOI: https://doi.org/10.1007/978-3-031-44505-7_29
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-44504-0
Online ISBN: 978-3-031-44505-7
eBook Packages: Computer ScienceComputer Science (R0)