- Prior probability is the probability available before we observe any new evidence.
- Posterior probability is the probability available after we observe evidence.
For example, the probability that someone has cancer is given by . This is the prior probability. Once we are told that this person received a positive test, we can use Bayes’ theorem to compute . This is the posterior probability.
This can yield surprising results; for example, in our example, the probability that someone has cancer when they have a positive test, , is still very low (23% in DLFC book), even if the test is reasonably accurate. This is because of the low prior probability of having cancer; although the test provides strong evidence of cancer, we have to combine it with the prior probability using Bayes’ theorem to arrive at the correct posterior probability.