Bayes Theorem

Bayes theorem is a part of probability theory that relates conditional probabilities and can be used to update beliefs (expressed as probabilities) based on new evidence. This is used in a number of approaches in AI including Bayesian network. To be precise Bayes theorem says:
      P( X | E )   =   ( P( E | X ) P(X) ) / sum ( P( E | X_i ) P(X_i) )
where X is some unknown state (such as "it is raining") the sum is taken over all possible states X_i (e.g. "it is dry", "it is snowing") and E is some other observable (often interpreted as evidence, such as "there's a wet umbrella in the hall-stand"). It effectively allows one to turn conditional probabiliies round.

When being used to update beliefs using evidence the probabilities P(X_i) are known as the prior distribution, whereas P(X_j|E) is the {{posterior distribution}.