Prior

Belief is a matter of degree.

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Bayes’ rule

An observation can shift a belief without settling the question.

Prior / A closer look

01 / The idea

Bayes’ rule

An observation can shift a belief without settling the question.

Start with a probability. Ask how well the observation fits each possibility. Update in proportion to the evidence.

Reading the symbols

H · E
H is the hypothesis; E is the observation. The bar | means “given”.
P(H)
Prior. How probable the hypothesis is before this observation.
P(E | H)
Likelihood. How probable this observation would be if the hypothesis were true.
P(E)
How probable the observation is overall, accounting for the alternatives.
P(H | E)
Posterior. The probability after incorporating the observation.

Illustrative example

Two coins, one observation

Choose coin A or coin B, then observe one head. B is fair: its chance of heads is 50%. H means “coin A was chosen”.

50%
60%
BeforeAfter0%50%100%
50% → 54.5%Probability that coin A was chosen

P(E) = 55% · the overall chance of a head, weighted across both coins.

A head can shift the probability without identifying the coin with certainty. Equal likelihoods leave the prior unchanged.

Keep in mind

The size and direction of the update depend on how differently the possibilities explain the observation. A small sample can matter; its reliability and context still matter too.

Bayesian Reasoning and Methods · Bayes’ ruleRead the source (opens a new tab) ↗