# Bayesian probability

Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of statements, or to the degree of belief of rational agents in the truth of statements. It is opposed to frequentism, which rejects degree-of-belief interpretations of mathematical probability, and assigns probabilities instead to random events according to their relative frequencies of occurrence. Whereas a frequentist might assign probability 1/2 to the event of getting a head when a coin is tossed (but only if the frequentist knows that that is the relative frequency) a Bayesian might assign probability 1/2 (or some other figure) to personal belief in the proposition that there was life on Mars a billion years ago, without intending that assignment to assert anything about any relative frequency.

The terms subjectivism, subjective probability, personal probability, epistemic probability and logical probability are used to describe what Bayesians believe in. Not all of these terms are synonymous, however.

Advocates of logical probability would like to codify techniques whereby if two people have the same information relevant to the truth of an uncertain proposition, then they would assign the same probability. No one has any idea how to do that except in simple cases, and then the validity of proposed methods is subject to philosophical controversy. The proponents of this view include Sir Harold Jeffreys, Richard Threlkeld Cox, and Edwin Jaynes. Its critics challenge the suggestion that it is possible or necessary in the absence of information to start with an objective prior belief which would be acceptable to all.

Subjective probability is supposed to measure how sure an individual is of an uncertain proposition.

The Bayesian approach is in contrast to frequency probability where probability is held to be derived from observed or imagined frequency distributions or proportions of populations. The difference has many implications for the methods by which statistics is practiced when following one model or the other, and also for the way in which conclusions are expressed. When comparing two hypotheses and using some information, frequency methods would typically result in the rejection or non-rejection of the original hypothesis with a particular degree of confidence, while Bayesian methods would suggest that one hypothesis was more probable than the other or that the expected loss associated with one was less than the expected loss of the other.

Bayesianism is proposed as a model of the scientific method. It is claimed that updating probabilities via Bayes' theorem is similar to the scientific method, in which one starts with an initial set of beliefs about the relative plausiblity of various hypotheses, collects new information (for example by conducting an experiment), and adjusts the original set of beliefs in the light of the new information to produce a more refined set of beliefs of the plausibility of the different hypotheses. See Bayesian inference for more information in this regard.

## History of Bayesian probability

Bayesianism is named after Thomas Bayes, who proved a special case of Bayes' theorem. This theorem is often used to update the plausibility of a given statement in light of new evidence. Laplace rediscovered the theorem and put it to good use in solving problems in celestial mechanics, medical statistics and, by some accounts, even jurisprudence.

For instance, Laplace estimated the mass of Saturn, given orbital data that were available to him from various astronomical observations. He presented the result together with an indication of its uncertainty, stating it like this: `It is a bet of 11000 to 1 that the error in this result is not within 1/100th of its value'. He would have won the bet, as another 150 years' accumulation of data has changed the estimate by only 0.63%. According to the frequency probability definition, however, the laws of probability are not applicable to this problem. This is because the mass of Saturn is a constant and not a random variable, therefore, it has no frequency distribution and so the laws of probability cannot be used.

Bayesianism has been promoted by L. J. Savage, Bruno de Finetti, Harold Jeffreys, Richard T. Cox, Edwin Jaynes, Frank P. Ramsey, John Maynard Keynes, B.O. Koopman, and others. They created the idea of defining rational belief as an abstraction of betting behavior subject to the constraint that one doesn't want to be inconsistent in his behavior. A series of critiques of statistical methods was based on this concept and formed the basis of debate from the 1950s and statisticians remain divided on the issue.

## Applications of Bayesian probability

Today, there are a variety of applications of personal probability that have gained wide acceptance. Some schools of thought emphasise Cox's theorem and Jaynes' principle of maximum entropy as cornerstones of the theory, while others may claim that Bayesian methods are more general and give better results in practice than frequency probability. See Bayesian inference for applications and Bayes' Theorem for the mathematics.