What you do to your beliefs, opinions and cognitive structure when new evidence comes along.

Cox's theorem says, roughly, that if your beliefs at any given time take the form of an assignment of a numerical "plausibility score" to every proposition, and if they satisfy a few plausible axioms, then your plausibilities must effectively be probabilities obeying the usual laws of probability theory, and your updating procedure must be the one implied by Bayes' theorem.

See also

Evidence, Conservation of expected evidence
Belief
Beliefs require observations
Bayes' theorem
Updateless decision theory