Bayes’ theorem

EstablishedMathematicsEpistemicsBiologyAI / ML

\htmlClass{sjt-post}{P(H \mid E)} = \frac{\htmlClass{sjt-like}{P(E \mid H)}\,\htmlClass{sjt-prior}{P(H)}}{\htmlClass{sjt-evid}{P(E)}}

Tap a symbol above, or choose one here:

P(H\mid E)

PosteriorRosetta entry →

Meaning

Probability the hypothesis is true after seeing the evidence.

Its job in this equation

The output: updated belief.

Picture it

Your revised opinion.

Read it, progressively

1. Gist

Update beliefs in proportion to how well each hypothesis predicted what you saw, without forgetting how plausible it was to begin with.

What it does not establish

The math is exact; the inputs are not. A posterior is only as good as its prior and likelihood model. It also does not tell you which hypotheses to consider, and leaving the true one out of the list cannot be fixed by updating.

Play with it

Bayesian updating

■ H true, positive (9)■ H false, positive (89)■ H true, missed (1)■ H false, negative (901)

Prior

P(H)

1.0%Sensitivity

P(E\mid H)

90%False-positive rate

P(E\mid \neg H)

9.0%

Posterior

P(H\mid E)

9.2%

Bayes factor (evidence strength)×10.0

Prior → posterior1.0% → 9.2%

Bayes’ rule, $P(H\mid E) = \dfrac{P(E\mid H)\,P(H)}{P(E)}$ , counted out. Among everyone who tests positive (filled squares), what fraction actually has $H$ ? When the prior is small, false positives from the large “not-H” group can swamp the true ones. Read “H” as “this research hypothesis is true” and the same arithmetic explains why surprising single studies often fail to replicate.

Loading interactive…

§ Module 4: Probability, Entropy, and Information Next equation: Softmax with temperature →