Belief that the use of induction in science can be rationally justified by appeal to Bayes's Theorem (Thomas Bayes, 1702-1761)).
This says that the probability of one proposition, given another, equals the probability of the second, given the first, multiplied by the prior probability of the first (that is the probability it already has, irrespective of the second) and divided by that of the second. The first proposition will be the one we are interested in, while the second will be some piece of evidence; and according to the theorem this will support the first proposition in proportion as its own prior probability is low, which suits our intuitions.
However, though the theorem itself is undisputed, its usefulness depends on our assigning suitable prior probabilities, with numerical values, to the two propositions. This raises problems.
'Bayesianism' is also sometimes used of any conception of rationality based on maximizing expected utilities, which links it to subjectivist theories of probability.