Concept
Updating beliefs with evidence — from an 18th century reverend to AI today.
Understand it in one breath
How should the probability of a hypothesis change after new evidence is observed? The theorem developed from Bayes's posthumously published essay and Laplace's later generalization. It reverses conditional probabilities and supports medical testing, spam classification, and scientific inference, but a poor prior or model can still produce a poor conclusion.
At a glance
Step | Calculation | Value |
|---|---|---|
Assumptions | Prevalence 1%; sensitivity 99%; specificity 99% | |
True positives among patients | P(+|disease) × P(disease) = 0.99 × 0.01 | 0.0099 |
False positives among non-patients | P(+|¬disease) × P(¬disease) = 0.01 × 0.99 | 0.0099 |
All positive results | 0.0099 + 0.0099 | 0.0198 |
Probability of disease given a positive result | 0.0099 / 0.0198 | 50% — contrary to intuition! |
Under these assumptions, the probability of disease after a positive result is 50%. Even with equal sensitivity and specificity, low prevalence can make false positives as numerous as true positives. Real interpretation requires test-specific performance and patient context.
Key formula
Worked examples
- 1
Q.A disease has 1% prevalence. A test has 99% sensitivity and 99% specificity. After a positive result, what is the probability of actually having the disease?
Key moments
Bayes’s work is published posthumously
After Thomas Bayes’s death, Richard Price presented his essay on inverse probability. It attracted limited attention at the time.
Laplace — a general theory
Laplace developed Bayesian reasoning into a broad method for inference from effects to causes across astronomy and science.
A revolution in medical diagnosis
Questions such as the meaning of a positive HIV test made base rates and posterior probability central to clear medical decision-making.
Bayesian machine learning
Bayesian neural networks and variational inference grew alongside modern machine learning, providing tools for representing uncertainty in predictions.
Modern applications
Medical diagnosis, spam filters, A/B tests, Bayesian networks, earthquake forecasting, and the evaluation of legal evidence.
Beyond MathVoyage
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