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The New England Journal of Statistics in Data Science


Fairness and Randomness in Machine Learning: Statistical Independence and Relativization
Pub. online: 19 November 2024      Type: Machine Learning And Data Mining      Open accessOpen Access
Published
19 November 2024

Abstract

Fair Machine Learning endeavors to prevent unfairness arising in the context of machine learning applications embedded in society. To this end, several mathematical fairness notions have been proposed. The most known and used notions turn out to be expressed in terms of statistical independence, which is taken to be a primitive and unambiguous notion. However, two choices remain (and are largely unexamined to date): what exactly is the meaning of statistical independence and what are the groups to which we ought to be fair? We answer both questions by leveraging Richard Von Mises’ theory of probability, which starts with data, and then builds the machinery of probability from the ground up. In particular, his theory places a relative definition of randomness as statistical independence at the center of statistical modelling. Much in contrast to the classically used, absolute i.i.d.-randomness, which turns out to be “orthogonal” to his conception. We show how Von Mises’ frequential modeling approach fits well to the problem of fair machine learning and show how his theory (suitably interpreted) demonstrates the equivalence between the contestability of the choice of groups in the fairness criterion and the contestability of the choice of relative randomness. We thus conclude that the problem of being fair in machine learning is precisely as hard as the problem of defining what is meant by being random. In both cases there is a consequential choice, yet there is no universal “right” choice possible.

References

[1] 
Abraham, K. S. Efficiency and fairness in insurance risk classification. Virginia Law Review 71(3) 403–451 (1985).
[2] 
Altman, A. Discrimination. In (E. N. Zalta, ed.) The Stanford Encyclopedia of Philosophy (Summerition), 2020th ed. Stanford University (2020). https://plato.stanford.edu/entries/discrimination/
[3] 
Ambos-Spies, K., Mayordomo, E., Wang, Y. and Zheng, X. Resource-bounded balanced genericity, stochasticity and weak randomness. In Annual Symposium on Theoretical Aspects of Computer Science 61–74. Springer (1996). MR1462086
[4] 
Bandyopadhyay, P. S. and Forster, M. R. Philosophy of statistics: An introduction. In (P. S. Bandyopadhyay and M. R. Forster, eds.) Philosophy of Statistics 7. Elsevier (2011). https://doi.org/10.1016/B978-0-444-51862-0.50001-0. MR3295937
[5] 
Solon Barocas, Hardt, M. and Narayanan, A. Fairness and Machine Learning: Limitations and Opportunities (2019). http://www.fairmlbook.org
[6] 
Battersby, M. The rhetoric of numbers: Statistical inference as argumentation. In OSSA Conference Archive (2003).
[7] 
Bennett, D. Defining randomness. In (P. S. Bandyopadhyay and M. R. Forster, eds.) Philosophy of Statistics 7. Elsevier (2011). https://doi.org/10.1016/B978-0-444-51862-0.50001-0. MR3295937
[8] 
Berkovitz, J., Frigg, R. and Kronz, F. The ergodic hierarchy, randomness and hamiltonian chaos. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 37(4) 661–691 (2006). https://doi.org/10.1016/j.shpsb.2006.02.003. MR2344115
[9] 
Biddle, D. Adverse Impact and Test Validation: A Practitioner’s Guide to Valid and Defensible Employment Testing. Gower Publishing, Ltd. (2006).
[10] 
Bienvenu, L., Shafer, G. and Shen, A. On the history of martingales in the study of randomness. Electronic Journal for History of Probability and Statistics 5(1) 1–40 (2009). MR2520666
[11] 
Binns, R. Fairness in machine learning: Lessons from political philosophy. In Conference on Fairness, Accountability and Transparency 149–159. PMLR (2018).
[12] 
Binns, R. On the apparent conflict between individual and group fairness. In Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency 514–524 (2020).
[13] 
Blum, N. Einführung in Formale Sprachen, Berechenbarkeit, Informations-und Lerntheorie. Oldenbourg Wissenschaftsverlag (2009).
[14] 
Broome, J. Selecting people randomly. Ethics 95(1) 38–55 (1984).
[15] 
Buss, S. and Minnes, M. Probabilistic algorithmic randomness. The Journal of Symbolic Logic 78(2) 579–601 (2013). MR3145197
[16] 
Toon, C. and Verwer, S. Three naive Bayes approaches for discrimination-free classification. Data Mining and Knowledge Discovery 21(2) 277–292 (2010). https://doi.org/10.1007/s10618-010-0190-x. MR2720507
[17] 
Campbell, D. T. Common fate, similarity, and other indices of the status of aggregates of persons as social entities. Behavioral Science 3(1) 14–25 (1958).
[18] 
Casella, G. and Berger, R. L. Statistical Inference 2nd ed. Duxbury Advanced Series (2002). MR1051420
[19] 
Castano, E., Yzerbyt, V., Paladino, M.-P. and Sacchi, S. I belong, therefore, I exist: Ingroup identification, ingroup entitativity, and ingroup bias. Personality and Social Psychology Bulletin 28(2) 135–143 (2002).
[20] 
Chaitin, G. J. On the length of programs for computing finite binary sequences. Journal of the ACM 13(4) 547–569 (1966). https://doi.org/10.1145/321356.321363. MR0210520
[21] 
Chichilnisky, G. The foundations of statistics with black swans. Mathematical Social Sciences 59(2) 184–192 (2010). https://doi.org/10.1016/j.mathsocsci.2009.09.007. MR2650318
[22] 
Chouldechova, A. Fair prediction with disparate impact: A study of bias in recidivism prediction instruments. Big Data 5(2) 153–163 (2017).
[23] 
Church, A. On the concept of a random sequence. Bulletin of the American Mathematical Society 46(2) 130–135 (1940). https://doi.org/10.1090/S0002-9904-1940-07154-X. MR0000911
[24] 
Cook, D. C. The concept of independence in accounting. In Federal Securities Law and Accounting 1933–1970: Selected Addresses 198–222. Routledge (2020).
[25] 
Dawid, P. On individual risk. Synthese 194(9) 3445–3474 (2017). https://doi.org/10.1007/s11229-015-0953-4. MR3704899
[26] 
De Cooman, G. and De Bock, J. Randomness is inherently imprecise. International Journal of Approximate Reasoning 141. 28–68 (2022). https://doi.org/10.1016/j.ijar.2021.06.018. MR4364895
[27] 
De Finetti, B. Funzione caratteristica di un fenomeno aleatorio. In Atti del Congresso Internazionale dei Matematici: Bologna del 3 al 10 de settembre di 1928 179–190 (1929).
[28] 
De Moivre, A. The Doctrine of Chances: A Method of Calculating the Probabilities of Events in Play 2nd ed. Frank Cass and Company Limited (1738/1967). MR0231695
[29] 
Deutsch, D. The Beginning of Infinity: Explanations that Transform the World. Penguin (2011). MR2984795
[30] 
Devroye, L., Györfi, L. and Lugosi, G. A Probabilistic Theory of Pattern Recognition. Springer (1996). https://doi.org/10.1007/978-1-4612-0711-5. MR1383093
[31] 
Downey, R. G. and Hirschfeldt, D. R. Algorithmic Randomness and Complexity. Springer (2010). https://doi.org/10.1007/978-0-387-68441-3. MR2732288
[32] 
Durrett, R. Probability: Theory and Examples 5th ed. Cambridge University Press (2019). https://doi.org/10.1017/9781108591034. MR3930614
[33] 
Dwork, C., Hardt, M., Pitassi, T., Reingold, O. and Zemel, R. Fairness through awareness. In Proceedings of the 3rd Innovations in Theoretical Computer Science Conference – ITCS 2012 214–226 (2012). MR3388391
[34] 
Dwork, C., Lee, D., Lin, H. and Tankala, P. From pseudorandomness to multi-group fairness and back. In The Thirty Sixth Annual Conference on Learning Theory 3566–3614. PMLR (2023).
[35] 
Eagle, A. Chance versus Randomness. In (E. N. Zalta, ed.) The Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University, Spring 2021 edition, 2021.
[36] 
Fine, T. L. Theories of Probability: An Examination of Foundations. Academic Press (1973). MR0433529
[37] 
Fröhlich, C., Derr, R. and Williamson, R. C. Towards a strictly frequentist theory of imprecise probability. In (E. Miranda, I. Montes, E. Quaeghebeur and B. Vantaggi, eds.) Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications. Proceedings of Machine Learning Research 215 230–240. PMLR (2023). MR4663313
[38] 
Frongillo, R. M. and Nobel, A. B. Memoryless sequences for general losses. J. Mach. Learn. Res. 21 80 (2020). MR4119148
[39] 
Gallie, W. B. Essentially contested concepts. In Proceedings of the Aristotelian society 56 167–198 (1955).
[40] 
Gammerman, A. and Vovk, V. Hedging predictions in machine learning. The Computer Journal 50(2) 151–163 (2007).
[41] 
Gillies, D. An Objective Theory of Probability. Routledge (2010) (First published 1973).
[42] 
Grädel, E. and Väänänen, J. Dependence and independence. Studia Logica 101(2) 399–410 (2013). https://doi.org/10.1007/s11225-013-9479-2. MR3038039
[43] 
Gudder, S. P. Quantum probability spaces. Proceedings of the American Mathematical Society 21(2) 296–302 (1969). https://doi.org/10.2307/2036988. MR0243793
[44] 
Gudder, S. P. Stochastic Methods in Quantum Mechanics. Dover Publications Inc. (1979). MR0543489
[45] 
Hájek, A. Fifteen arguments against hypothetical frequentism. Erkenntnis 70(2) 211–235 (2009). https://doi.org/10.1007/s10670-009-9154-1. MR2481794
[46] 
Hardt, M., Price, E. and Srebro, N. Equality of opportunity in supervised learning. In Advances in Neural Information Processing Systems 3323–3331 (2016).
[47] 
Hertweck, C., Heitz, C. and Loi, M. On the moral justification of statistical parity. In Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency 747–757 (2021).
[48] 
Hogg, M. A. and Abrams, D. Social Identifications: A Social Psychology of Intergroup Relations and Group Processes. Routledge (1998).
[49] 
Hu, L. and Kohler-Hausmann, I. What’s sex got to do with fair machine learning? arXiv preprint arXiv:2006.01770 (2020).
[50] 
Humphreys, P. W. Randomness, independence, and hypotheses. Synthese 36 415–426 (1977). https://doi.org/10.1007/BF00486105. MR0517129
[51] 
Ivanenko, V. I. Decision Systems and Nonstochastic Randomness. Springer (2010). https://doi.org/10.1007/978-1-4419-5548-7. MR2606231
[52] 
Kamishima, T., Akaho, S., Asoh, H. and Sakuma, J. Fairness-aware classifier with prejudice remover regularizer. In Proceedings of the 2012th European Conference on Machine Learning and Knowledge Discovery in Databases-Volume Part II ECMLPKDD’12: 35–50 (2012).
[53] 
Kilbertus, N., Rojas Carulla, M., Parascandolo, G., Hardt, M., Janzing, D. and Schölkopf, B. Avoiding discrimination through causal reasoning. In Advances in Neural Information Processing Systems 30 (2017).
[54] 
Kleinberg, J., Mullainathan, S. and Raghavan, M. Inherent trade-offs in the fair determination of risk scores. In Leibniz International Proceedings in Informatics, LIPIcs 67 1–22 (2017). MR3754967
[55] 
Kolmogorov, A. N. Grundbegriffe de Wahrscheinlichkeitsrechnung. Springer (1933). MR0362415
[56] 
Kolmogorov, A. N. Foundations of the Theory of Probability: Second English Edition. Chelsea Publishing Company (1956). MR0079843
[57] 
Kolmogorov, A. N. Three approaches to the definition of the concept “quantity of information”. Problemy Peredachi Informatsii 1(1) 3–11 (1965). MR0184801
[58] 
Lafferty, J. and Wasserman, L. Challenges in statistical machine learning. Statistica Sinica 16(2) 307 (2006). MR2267237
[59] 
Levin, L. A. The concept of random sequence. Soviet Mathematics Doklady 14. 1413–1416 (1973). MR0366096
[60] 
Levin, L. A. A concept of independence with applications in various fields of mathematics. Technical Report MIT/LCS/TR-235, MIT, Laboratory for Computer Science, April 1980.
[61] 
Lippert-Rasmussen, K. The badness of discrimination. Ethical Theory and Moral Practice 9(2) 167–185 (2006).
[62] 
Vitali, L. Analysis 1 Lecture Notes 2013/2014. University of Bristol, 2014.
[63] 
Lohaus, M., Perrot, M. and Von Luxburg, U. Too relaxed to be fair. In International Conference on Machine Learning 6360–6369. PMLR (2020).
[64] 
Loi, M., Herlitz, A. and Heidari, H. A philosophical theory of fairness for prediction-based decisions. Technical report, Politecnico di Milano, 2019. https://ssrn.com/abstract=3450300
[65] 
Martin-Löf, P. The definition of random sequences. Information and Control 9(6) 602–619 (1966). MR0223179
[66] 
McGarty, C. Categorization in Social Psychology. Sage Publications (1999).
[67] 
Menon, A. K. and Williamson, R. C. The cost of fairness in binary classification. In Conference on Fairness, Accountability and Transparency 107–118. PMLR (2018).
[68] 
Muchnik, A. A., Semenov, A. L. and Uspensky, V. A. Mathematical metaphysics of randomness. Theoretical Computer Science 207(2) 263–317 (1998). https://doi.org/10.1016/S0304-3975(98)00069-3. MR1643438
[69] 
Murchison, B. C. The concept of independence in public law. Emory Law Journal 41. 961 (1992).
[70] 
Narens, L. An introduction to lattice based probability theories. Journal of Mathematical Psychology 74. 66–81 (2016). https://doi.org/10.1016/j.jmp.2016.04.013. MR3552130
[71] 
Nathanson, M. B. Elementary Methods in Number Theory 195. Springer (2008). MR1732941
[72] 
Naylor, A. W. On decomposition theory: generalized dependence. IEEE Transactions on Systems, Man, and Cybernetics 11(10) 699–713 (1981). https://doi.org/10.1109/TSMC.1981.4308590. MR0641948
[73] 
Donald and Ornstein, S. Ergodic theory, randomness, and “chaos”. Science 243(4888) 182–187 (1989). https://doi.org/10.1126/science.243.4888.182. MR0981173
[74] 
Parker, J. M. Randomness and legitimacy in selecting democratic representatives. The University of Texas at Austin, 2011.
[75] 
Passi, S. and Barocas, S. Problem formulation and fairness. In Proceedings of the Conference on Fairness, Accountability, and Transparency 39–48 (2019).
[76] 
Popper, K. The Logic of Scientific Discovery. Routledge (2010). MR1195353
[77] 
Porter, C. P. Mathematical and philosophical perspectives on algorithmic randomness. PhD thesis, University of Notre Dame, 2012. MR3217940
[78] 
Pták, P. Concrete quantum logics. International Journal of Theoretical Physics 39(3) 827–837 (2000). https://doi.org/10.1023/A:1003626929648. MR1792201
[79] 
Putnam, H. The Meaning of the Concept of Probability in Application to Finite Sequences (Routledge Revivals). Garland Publishing (1990).
[80] 
Raghavan, M., Barocas, S., Kleinberg, J. and Levy, K. Mitigating bias in algorithmic hiring: Evaluating claims and practices. In Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency 469–481 (2020).
[81] 
Bhaskara Rao, K. P. S. and Bhaskara Rao, M. Theory of Charges: A Study of Finitely Additive Measures. Academic Press (1983). MR0751777
[82] 
Rawls, J. A Theory of Justice. The Belknap Press of Harvard University Press (1971).
[83] 
Räz, T. Group fairness: Independence revisited. In Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency 129–137 (2021). MR4297512
[84] 
Scantamburlo, T. Non-empirical problems in fair machine learning. Ethics and Information Technology 23(4) 703–712 (2021).
[85] 
Schnorr, C.-P. Zufälligkeit und Wahrscheinlichkeit: eine algorithmische Begründung der Wahrscheinlichkeitstheorie. Lecture Notes in Mathematics 218. Springer (1971). MR0414225
[86] 
Schnorr, C.-P. The process complexity and effective random tests. In Proceedings of the Fourth Annual ACM Symposium on Theory of Computing 168–176 (1972). https://doi.org/10.1016/S0022-0000(73)80030-3. MR0325366
[87] 
Schnorr, C.-P. A survey of the theory of random sequences. In Basic Problems in Methodology and Linguistics 193–211. Springer (1977). MR0517133
[88] 
Schurz, G. and Leitgeb, H. Finitistic and frequentistic approximation of probability measures with or without σ-additivity. Studia Logica 89(2) 257–283 (2008). https://doi.org/10.1007/s11225-008-9128-3. MR2429951
[89] 
Shafer, G. Discussion on hedging predictions in machine learning by A. Gammerman and V. Vovk. The Computer Journal 50(2) 164–172 (2007). https://doi.org/10.1093/comjnl/bxl066
[90] 
Shafer, G. and Vovk, V. Game-Theoretic Foundations for Probability and Finance 455. John Wiley & Sons (2019). https://doi.org/10.1002/0471249696. MR1852450
[91] 
Shalev-Shwartz, S. and Ben-David, S. Understanding Machine Learning: From Theory to Algorithms. Cambridge University Press (2014). MR3277164
[92] 
Simpson, J. and Weine, E. “independence, n”. In Oxford English Dictionary, 2nd ed. Oxford University Press (1989).
[93] 
Spohn, W. Stochastic independence, causal independence, and shieldability. Journal of Philosophical Logic 9(1) 73–99 (1980). https://doi.org/10.1007/BF00258078. MR0563250
[94] 
Steinwart, I., Hush, D. and Scovel, C. Learning from dependent observations. Journal of Multivariate Analysis 100(1) 175–194 (2009). https://doi.org/10.1016/j.jmva.2008.04.001. MR2460486
[95] 
Stone, P. The Luck of the Draw: The Role of Lotteries in Decision-Making. Oxford University Press (2011).
[96] 
Tadaki, K. An operational characterization of the notion of probability by algorithmic randomness. In Proceedings of the 37th Symposium on Information Theory and its Applications (SITA2014) 5 389–394 (2014).
[97] 
Tajfel, H. Social identity and intergroup behaviour. Social Science Information 13(2) 65–93 (1974).
[98] 
Tao, T. 275a, notes 2: Product measures and independence, October 2015. https://terrytao.wordpress.com/2015/10/12/275a-notes-2-product-measures-and-independence/
[99] 
Uspenskii, V. A., Semenov, A. L. and Shen, A. Kh. Can an individual sequence of zeros and ones be random? Russian Mathematical Surveys 45(1) 121 (1990). https://doi.org/10.1070/RM1990v045n01ABEH002321. MR1050929
[100] 
Van Lambalgen, M. Von Mises’ definition of random sequences reconsidered. The Journal of Symbolic Logic 52(3) 725–755 (1987). https://doi.org/10.2307/2274360. MR0902987
[101] 
Van Lambalgen, M. The axiomatization of randomness. The Journal of Symbolic Logic 55(3) 1143–1167 (1990). https://doi.org/10.2307/2274480. MR1071321
[102] 
Ville, J. Étude critique de la notion de collectif. Gauthier-Villars (1939). MR3533075
[103] 
Volchan, S. B. What is a random sequence? The American Mathematical Monthly 109(1) 46–63 (2002). https://doi.org/10.2307/2695767. MR1903512
[104] 
Von Collani, E. A note on the concept of independence. Economic Quality Control 21(1) 155–164 (2006).
[105] 
Von Mises, R. Grundlagen der Wahrscheinlichkeitsrechnung. Mathematische Zeitschrift 5(1) 52–99 (1919). https://doi.org/10.1007/BF01203155. MR1544374
[106] 
Von Mises, R. Probability, Statistics, and Truth. Dover Publications, Inc. (1981). MR0668875
[107] 
Von Mises, R. and Geiringer, H. Mathematical Theory of Probability and Statistics. Academic Press (1964). MR0178486
[108] 
Vovk, V. G. A logic of probability, with application to the foundations of statistics. Journal of the Royal Statistical Society: Series B (Methodological) 55(2) 317–341 (1993). MR1224399
[109] 
Wasserman, L. All of Statistics: A Concise Course in Statistical Inference. Springer (2004). https://doi.org/10.1007/978-0-387-21736-9. MR2055670
[110] 
Wick, M., Panda, S. and Tristan, J.-B. Unlocking fairness: a trade-off revisited. In Proceedings of the 37th International Conference on Machine Learning 32 (2020).
[111] 
Williamson, R. C. and Menon, A. Fairness risk measures. In International Conference on Machine Learning 6786–6797. PMLR (2019).

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© 2024 New England Statistical Society
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Keywords
Fairness in machine learning Richard von Mises Randomness Statistical independence

Funding
This work was funded in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy — EXC number 2064/1 — Project number 390727645; it was also supported by the German Federal Ministry of Education and Research (BMBF): Tübingen AI Center.

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