Bayesian probability

Named after the English mathematician Thomas Bayes Bayesian concept of probability (English Bayesianism ) interprets probability as a degree of personal conviction (english degree of belief ). He thus differs interpreted as relative frequency of the objectivist probability concepts such as the frequentist concept of probability, the probability.

The Bayesian probability term should not be confused with the equally going back to Thomas Bayes Bayestheorem, which is rich application in statistics.

Development of Bayesian probability concept

The Bayesian probability term is often used to remeasure the plausibility of a statement in light of new evidence. Pierre- Simon Laplace (1812 ) discovered this sentence later independently of Bayes and used it to solve problems in celestial mechanics, medical statistics and, according to some reports, even in the case law.

For example, Laplace estimated the mass of Saturn, based on existing astronomical observations of its orbit. He explained the results along with an indication of its uncertainty: "I bet 11,000 to 1 that the error in this result is not greater than 1/100 of its value. " ( Laplace had won his bet, and 150 years later had to be the result based on new data are corrected by only 0.37%. )

The Bayesian interpretation of probability was first beginning of the 20th century worked mainly in England. Leading minds were about Harold Jeffreys (1891-1989) and Frank Plumpton Ramsey ( 1903-1930 ). The latter developed an approach, which he could not pursue due to his early death, but was taken up in Italy regardless of Bruno de Finetti ( 1906-1985 ). The basic idea is, "reasonable estimates " (English rationally was ) as a generalization of betting strategies conceived: Given a set of information / measurements / data points, and sought an answer to the question, how much to bet on the correctness of his assessment or what odds you would give. ( The background is that you can then just bet a lot of money when one is aware of its assessment safe. This idea had great influence on the game theory). A series of pamphlets against ( frequentist ) statistical methods came from this basic idea are being discussed since the 1950s between the Bayesians and frequentists.

Formalization of the concept of probability

If one is willing to interpret probability as a "safety in the personal assessment of facts " (see above), so the question arises which logical properties of this probability must have, in order not to be contradictory. Significant contributions were made to do so by Richard Threlkeld Cox ( 1946). He calls the validity of the following principles:

Probability values

It turns out that the following rules for probability values ​​W ( H) shall apply:

This means:

• H or D: hypothesis H is true ( or event occurs H ) or the event D occurs.
• W ( H): The probability that hypothesis H is true or event H will occur.
• ! H: Non- H: the hypothesis H is not true or the event H does not occur.
• H, D H and D are both true or both, or enter one is true and the other occurs.
• W ( D | H): The probability that D hypothesis is true or enter event D in the case that H would or would occur true.

One can easily see that the probability values ​​must start at 0; otherwise something would be like a ' twice as high probability ' have no meaning.

From the above rules of probability values ​​other can be derived.

Practical significance in statistics

Such problems to still be able to adopt during the frequentist interpretation, the uncertainty is described there by means of the specially invented variable random variable. The Bayesian probability theory does not require such an auxiliary variable. Instead, it introduces the concept of a priori probability, the prior knowledge and assumptions of the observer summarized in a probability distribution. Representatives of the Bayesian approach see it as a great advantage of prior knowledge and a priori assumptions explicitly express in the model.