Bayes' Theorem
Rare disease + good test = little certainty
Summary
If a medical test comes back positive, what is the chance that you are ill? Bayes' theorem gives the answer, and it is often very different from intuition. Use the sliders to see why, for a rare disease and a good test, a positive result is usually wrong.
In depth
In 1763 (published after his death), Thomas Bayes gave a formula for conditional probability: how the probability of an event should be updated once another event has happened. The medical test example makes this concrete. A test's sensitivity is the rate at which it correctly identifies people who are ill, and its specificity the rate at which it correctly identifies people who are healthy. Even when both are high, if the disease is very rare (low prevalence) the number of false positives can be larger than the number of true positives; as a result, the probability that a positive test really means disease (the PPV, positive predictive value) is low. This counter-intuitive result matters greatly for medical decision-making, coronavirus screening programmes, the evaluation of evidence in court and machine learning (especially with imbalanced datasets).
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