{"course":{"slug":"machine-learning-foundations","title":"Machine Learning, Foundations","url":"https://addaly.com/learn/machine-learning-foundations","level":"intermediate","track":"start"},"lesson":{"slug":"the-impossibility-result","title":"Why you cannot have all three","url":"https://addaly.com/learn/machine-learning-foundations/the-impossibility-result","position":51,"of":92,"minutes":8,"answer":"When two groups have different base rates, calibration within groups and equal error rates across groups are mathematically incompatible, so fairness requires an explicit and documented choice of which to satisfy.","keyIdea":"When two groups have different base rates, calibration within groups and equal error rates across groups are mathematically incompatible, so fairness requires an explicit and documented choice of which to satisfy."},"body":{"format":"markdown","text":"## A theorem, not an opinion\n\nThe previous lesson gave three fairness criteria. This one gives the result that makes choosing between them unavoidable.\n\n> If two groups have different base rates for the outcome, then no classifier can simultaneously satisfy calibration within groups and equal false positive *and* false negative rates, except a perfect classifier, or one that has no predictive power at all.\n\nThis was proved independently by Kleinberg, Mullainathan and Raghavan, and by Chouldechova, both in 2016. It is a mathematical result. It does not depend on the algorithm, the data quality, the features, or how careful you are.\n\n## Seeing why\n\nThe intuition takes a paragraph.\n\nCalibration says: among everyone the model scored at 0.7, the same fraction are actually positive, in both groups. Now suppose group A has a 30% base rate and group B has 10%.\n\nTo be calibrated for group B, whose members are less likely to be positive on average, the model has to spread B's scores differently: a given score has to be reachable by the same fraction of true positives in each group. Doing that while also holding the false positive rate equal across groups is over-determined. The base rate appears in both requirements, pulling in different directions, and the only ways out are a model that gets everything right (no errors to distribute) or a model that predicts nothing (nothing to distribute).\n\nYou can verify this yourself with a spreadsheet: fix two base rates, fix a false positive rate, and try to satisfy calibration at every score band. It does not close.\n\n## COMPAS, again\n\nThis is exactly what happened in the recidivism argument. ProPublica showed unequal false positive rates. Northpointe showed calibration within groups. Base rates in the recorded data differed between groups.\n\nGiven that difference, both findings had to be true simultaneously. The theorem says so. Neither party was making an error, and neither could have satisfied the other without abandoning their own criterion.\n\nWhat the debate should have been about (and largely became, afterwards) is which criterion the situation demands, and whether the recorded base rate difference reflects the world or reflects policing patterns. The second question is not a statistical one at all.\n\n## What this means for your work\n\nThree consequences, and they are practical.\n\n**You must choose, and the choice is a value judgement.** There is no technically superior answer. There is a decision about which harm matters more in this context, and it belongs to the people accountable for the decision, not to the person writing the code. Your job is to make the trade-off legible so they can make it knowingly.\n\n**Write down the choice and its cost.** \"We equalised false negative rates across groups, which raised the false positive rate for group B from 8% to 13%. We chose this because a missed diagnosis is worse than an unnecessary follow-up.\" That sentence is what accountability looks like in this field.\n\n**Beware anyone selling a solution.** A tool that claims to make your model fair has picked a definition on your behalf. Ask which one. If the answer is not immediate and specific, the tool is doing something you cannot audit.\n\n## The escape that is not a trick\n\nThe theorem's condition is *different base rates*. If the base rates are equal, the conflict disappears.\n\nSo the question worth asking is why they differ. Sometimes the difference is real. Often it is an artefact of measurement: the outcome you recorded is not the outcome you care about. Recorded rearrest is not reoffending; it is reoffending filtered through where police patrol. Recorded default is not inability to pay; it is inability to pay filtered through who was offered what terms.\n\nWhen the measured base rate difference comes substantially from the measurement, the right intervention is upstream: fix the label, or change what you are predicting. That is slower and more disruptive than adjusting a threshold, and it is the only move that addresses the cause rather than distributing the consequence.\n\n## Why this is in a foundations course\n\nBecause it is the clearest case in machine learning where the mathematics tells you something a technique cannot fix. Most of this course teaches methods. This lesson teaches a limit, and the limit is more durable than any method in the syllabus: it will still hold when every algorithm here has been replaced."},"check":{"question":"A team is told their model must be both calibrated within groups and have equal false positive rates, and the two groups have measurably different base rates. What is the correct response?","options":[{"text":"Collect more data and improve the model until both criteria are met, since the conflict comes from estimation error.","correct":false,"tempting":"Treats the incompatibility as a data-quantity problem; the result is structural and holds for any imperfect classifier however much data it saw."},{"text":"Apply post-processing threshold adjustment, which can satisfy both criteria simultaneously.","correct":false,"tempting":"Post-processing can equalise one chosen criterion, but it redistributes errors rather than escaping the theorem, so the other criterion still breaks."},{"text":"Explain that the two requirements are mathematically incompatible at unequal base rates, ask which harm the organisation considers worse, and document the choice and what it costs the other criterion.","correct":true},{"text":"Use demographic parity instead, since it avoids the conflict between the other two definitions.","correct":false,"tempting":"Parity is a third criterion with its own costs, and switching to it is itself the value judgement being avoided rather than a way around the result."}]},"prev":{"slug":"fairness-you-can-compute","title":"Fairness as arithmetic: the definitions and what each one demands","url":"https://addaly.com/learn/machine-learning-foundations/fairness-you-can-compute"},"next":{"slug":"slice-based-error-analysis","title":"The average is hiding the failure","url":"https://addaly.com/learn/machine-learning-foundations/slice-based-error-analysis"},"authorship":"Founder and series editor: Dr Mustafa Mun. Written by AI models to a written standard. Exam keys checked blind by a second model: 3,665 of 3,666 agreed. Academic review invited.","review":{"status":"Draft, open for academic review"},"cite":"Addaly (2026) 'Why you cannot have all three', in Machine Learning, Foundations. Edited by M. Mun. Addaly. Available at: https://addaly.com/learn/machine-learning-foundations/the-impossibility-result (Accessed: 13 October 2026).","licence":{"name":"CC BY-NC-SA 4.0","url":"https://creativecommons.org/licenses/by-nc-sa/4.0/"},"reader":{"schema":1,"title":"Why you cannot have all three","courseTitle":"Machine Learning, Foundations","module":{"index":5,"slug":"measuring-honestly","title":"Measuring a model honestly"},"answer":null,"keyIdea":"When two groups have different base rates, calibration within groups and equal error rates across groups are mathematically incompatible, so fairness requires an explicit and documented choice of which to satisfy.","hero":{"kind":"plate","image":null,"alt":null,"caption":null},"blocks":[{"t":"h2","id":"a-theorem-not-an-opinion","runs":[{"text":"A theorem, not an opinion"}]},{"t":"p","runs":[{"text":"The previous lesson gave three fairness criteria. 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There is a decision about which harm matters more in this context, and it belongs to the people accountable for the decision, not to the person writing the code. Your job is to make the trade-off legible so they can make it knowingly."}]},{"t":"p","runs":[{"text":"Write down the choice and its cost.","b":true},{"text":" “We equalised false negative rates across groups, which raised the false positive rate for group B from 8% to 13%. We chose this because a missed diagnosis is worse than an unnecessary follow-up.” That sentence is what accountability looks like in this field."}]},{"t":"p","runs":[{"text":"Beware anyone selling a solution.","b":true},{"text":" A tool that claims to make your model fair has picked a definition on your behalf. Ask which one. 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