Equations · Probability and statistics
Bayes' theorem
How to update the probability of a hypothesis A after seeing evidence B.
P(A|B) = P(B|A) P(A) / P(B)
Aethel says
P of A given B equals P of B given A, times P of A, over P of B
What it says
Start with how likely A was before the evidence, P(A), the prior. Multiply by how likely the evidence is if A is true, P(B|A), the likelihood. Divide by how likely the evidence is overall, P(B). The result is the posterior: how likely A is now.
It explains why a positive result from an accurate test for a rare condition can still mean the condition is unlikely: the prior is small.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| P(A|B) | P of A given B | the probability of A when B is known to be true |
| P(A) | P of A | the probability of A before seeing B, the prior |
| | | given | the condition that is known |
Where you meet it
Statistics, medical testing, spam filters and machine learning.
Questions
How do you say Bayes' theorem out loud?
P(A|B) = P(B|A) P(A) / P(B) is read: “P of A given B equals P of B given A, times P of A, over P of B.” That is exactly what Aethel says when it meets the formula in a book.
What does P(A|B) mean in Bayes' theorem?
P(A|B) (said “P of A given B”) is the probability of A when B is known to be true.
What does P(A) mean in Bayes' theorem?
P(A) (said “P of A”) is the probability of A before seeing B, the prior.
Can I listen to the equations in my own textbook like this?
Yes. Upload a PDF, EPUB or text file and Aethel reads it aloud with every equation spoken as maths, the way it reads this one, and highlights each sentence as it goes. You can start free.