Is probability a long-run frequency or a degree of belief?

What does it mean to say that the probability of heads is one half? One answer, frequentism, says that an event's probability is the limit of its relative frequency in infinitely many trials. Probabilities can in principle be found by a repeatable objective process, such as repeated sampling from the same population, and are thus ideally devoid of subjectivity. The other answer, Bayesian probability, says that, instead of frequency, probability is a reasonable expectation representing a state of knowledge, or the quantification of a personal belief.

The two answers disagree about what can have a probability at all. For the frequentist, a hypothesis is either true or false, so its probability of being correct is 0 or 1. In Bayesian statistics, a hypothesis can be assigned a probability between 0 and 1 if its truth value is uncertain, starting from a prior probability that is updated to a posterior probability in the light of new data. Feller put the frequentist position bluntly: there is no place in the system for speculations concerning the probability that the sun will rise tomorrow.

The history swung back and forth. Thomas Bayes never published his most famous accomplishment; Richard Price edited and published his notes after his death. Pierre-Simon Laplace introduced a general version of the theorem and applied it to celestial mechanics, medical statistics, reliability, and jurisprudence. After the 1920s, "inverse probability" was largely supplanted by methods called frequentist statistics. Then Harold Jeffreys' Theory of Probability (1939) played an important role in the revival of the Bayesian view, and in the 1980s Bayesian methods grew dramatically, mostly thanks to Markov chain Monte Carlo methods. Both schools are alive today.