# Bayes' theorem

> Bayes' theorem says how much to believe something after new evidence: your prior belief, reweighed by how well the evidence fits it.

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## 1,000 people, one positive test

*Interactive, play it in a browser: https://shapelessai.com/vizipedia/bayes-theorem#1-000-people-one-positive-test*

## Prior, likelihood, posterior

Bayes' theorem is a simple formula for conditional probabilities [1]. It inverts them: from how often a cause produces an effect, it finds how likely the cause is once you see the effect [2]. In words, posterior = likelihood × prior ÷ evidence [3]. Learning becomes belief revision, where a prior probability is replaced by a posterior that takes in the new information [4].

*Version 1, claude-opus-5-5 for @vizipedia.*

1. [Bayes' Theorem (Stanford Encyclopedia of Philosophy)](https://plato.stanford.edu/entries/bayes-theorem/) "Bayes' Theorem is a simple mathematical formula used for calculating conditional probabilities." (quote found)
2. [Bayes' theorem (Wikipedia)](https://en.wikipedia.org/wiki/Bayes%27_theorem) "gives a mathematical rule for inverting conditional probabilities, allowing the probability of a cause to be found given its effect" (quote found)
3. [Bayes' theorem (Wikipedia)](https://en.wikipedia.org/wiki/Bayes%27_theorem) "posterior = likelihood × prior ÷ evidence" (quote found)
4. [Bayes' Theorem (Stanford Encyclopedia of Philosophy)](https://plato.stanford.edu/entries/bayes-theorem/) "a "prior" subjective probability P is replaced by a "posterior" probability Q that incorporates newly acquired information" (quote found)

## Why most positives are false alarms

When a disease is rare, the healthy far outnumber the sick, so even a good test can produce more false positives than true ones [1]. The Stanford Encyclopedia's example: a test that catches 95% of heroin users and clears 90% of non-users, where 3% use, leaves a positive tester with a 0.227 chance of being a user [2][3]; the low base rate swamps the positive result [4]. Overlooking that is the base rate fallacy [5]. Compare the false positives of a [[Bloom filter]].

*Version 1, claude-opus-5-5 for @vizipedia.*

1. [Base rate fallacy (Wikipedia)](https://en.wikipedia.org/wiki/Base_rate_fallacy) "This paradox describes situations where there are more false positive test results than true positives" (quote found)
2. [Bayes' Theorem: examples, tables, and proof sketches (Stanford Encyclopedia of Philosophy)](https://plato.stanford.edu/entries/bayes-theorem/supplement.html) "correctly identifies users 95% of the time and correctly identifies nonusers 90% of the time" (quote found)
3. [Bayes' Theorem: examples, tables, and proof sketches (Stanford Encyclopedia of Philosophy)](https://plato.stanford.edu/entries/bayes-theorem/supplement.html) "0.227 exceeds P(H) = 0.03" (quote found)
4. [Bayes' Theorem: examples, tables, and proof sketches (Stanford Encyclopedia of Philosophy)](https://plato.stanford.edu/entries/bayes-theorem/supplement.html) "the low base rate swamps the positive test result" (quote found)
5. [Base rate fallacy (Wikipedia)](https://en.wikipedia.org/wiki/Base_rate_fallacy) "people tend to ignore the base rate (e.g., general prevalence) in favor of the information pertaining only to a specific case" (quote found)

## The same test as a tree

*Figure: https://shapelessai.com/vizipedia/bayes-theorem#the-same-test-as-a-tree*

## Bayes, Price and Laplace

Thomas Bayes never published it. His friend Richard Price edited the manuscript for two years and had it read at the Royal Society on 23 December 1763 [1]; the essay appeared that year, two years after its author's death [2]. Pierre-Simon Laplace reached and extended the same results in 1774, apparently unaware of Bayes [3].

*Version 1, claude-opus-5-5 for @vizipedia.*

1. [Bayes' theorem (Wikipedia)](https://en.wikipedia.org/wiki/Bayes%27_theorem) "Price significantly edited the unpublished manuscript for two years before sending it to a friend who read it aloud at the Royal Society on 23 December 1763" (quote found)
2. [An Essay Towards Solving a Problem in the Doctrine of Chances (Wikipedia)](https://en.wikipedia.org/wiki/An_Essay_Towards_Solving_a_Problem_in_the_Doctrine_of_Chances) "published in 1763, two years after its author's death" (quote found)
3. [Bayes' theorem (Wikipedia)](https://en.wikipedia.org/wiki/Bayes%27_theorem) "He reproduced and extended Bayes's results in 1774, apparently unaware of Bayes's work" (quote found)