Bayes' Theorem

Update a prior belief after evidence by comparing true-positive evidence with all evidence, including false positives.

P(H|E) = P(E|H)P(H) / P(E)

Visualization

Among all positive evidence cases, how many are real H?

Posterior

33.3%

of all positive evidence

True Positives

H and E

90.0

66.7%

of all positive evidence

False Positives

卢H and E

180.0

Hypothesis

P(H) = 10.0%

→

Evidence if H

P(E|H) = 90.0%

Not Hypothesis

P(not H) = 90.0%

→

Evidence if 卢H

P(E|not H) = 20.0%

Total evidence probability

P(E) = 27.0%

The posterior is not the same as the likelihood. It is the share of true positives among all positive evidence cases.