Departure question
Read signals from chancePort 2 of 11Travel from dice and insurance to data, noise, and learning algorithms—measuring uncertainty and turning it into decisions.
This voyage is an editorial path for understanding, not a claim of direct historical influence or sole invention.
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.
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.
Updating beliefs with evidence — from an 18th century reverend to AI today.
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?
Ports in time
Follow the scenes to see problems, notation, standards of proof, and applications changing across different times and places.
After Thomas Bayes’s death, Richard Price presented his essay on inverse probability. It attracted limited attention at the time.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearLaplace developed Bayesian reasoning into a broad method for inference from effects to causes across astronomy and science.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearQuestions such as the meaning of a positive HIV test made base rates and posterior probability central to clear medical decision-making.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearBayesian neural networks and variational inference grew alongside modern machine learning, providing tools for representing uncertainty in predictions.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearMedical diagnosis, spam filters, A/B tests, Bayesian networks, earthquake forecasting, and the evaluation of legal evidence.
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Number lenses
These numbers are editorial lenses for the voyage, not required prerequisites.
Travel from dice and insurance to data, noise, and learning algorithms—measuring uncertainty and turning it into decisions.
Open the number voyage
Travel from dice and insurance to data, noise, and learning algorithms—measuring uncertainty and turning it into decisions.
Open the number voyage
Travel from dice and insurance to data, noise, and learning algorithms—measuring uncertainty and turning it into decisions.
Open the number voyage
Concept genealogy
Concepts arriving from before
Current port
Bayes' Theorem
Concepts opened from here
Only direct editorial links are shown; this is not a complete learning order or historical influence line.
