## Bayes theorem

This theorem deals with the impact of new information on the revision of probability estimates, and provides a normative model to assess how well people use empirical information to update the probability that a hypothesis is true.
*P*(*H*|*O*) = *P*(*H*) x *P*(*O*|*H*) / [ *P*(*H*) x *P*(*O*|*H*) + *P*(*nonH*) x *P*(*O*|*nonH*) ]

Bayes's theorem tells us that the probability that a hypothesis is true given that we
have made some observation (called the "posterior odds") *P(H|O)* is a function of:

*P(H)* = The probability you would have assigned to the hypothesis before you made the observation, called the "prior probability" of the hypothesis.

*P(O|H)* = The probability the observation would occur if the hypothesis were true.

*P(nonH)* = The prior probability the hypothesisis not true, *1-P(H)*.

*P(O|nonH)* = The probability the event would have occured even if the hypothesis were not true.

For example, when the baserates of women having breast cancer and having no breast
cancer are known to be 1% and 99%, respectively, and the hit rate is given as
*P*(positive mammography/ breast cancer) = 80 %, applying the Bayes theorem leads to
a normative prediction as low as *P*(breast cancer/ positive mammography) = 7.8%.
That means that the probability that a woman who has a positive mammography actually
has breast cancer is less than 8%. Studies show
(e.g. Gigerenzer & Hoffrage, 1995)
that subjective estimates clearly exceed the normative prediction and are often very
close to the hit rate (80% in the example).

**See also:**
Base-rate fallacy

**Literature:**
Bayes (1763/64),
Manstead & Hewstone (1995), p. 85,
Gigerenzer & Hoffrage (1995)