Bruno de Finetti

Bruno de Finetti ( born June 13, 1906 in Innsbruck, † 20 July 1985 in Rome) was an Italian mathematician. His most important results are to settle in statistics and probability theory.

Life

De Finetti came as Bruno Johannes Leonhard Maria von Finetti to the world. His grandfather, Giovanno Ritter von Finetti, was a building contractor from Trieste, which belonged at this time to Austria, and participated in the construction of the Arlberg railway. Therefore, the family moved to Innsbruck. De Finettis Gualtiero father took over the company until 1910 he got a reputation as Planning Director to Trieste. Shortly after his father died, and the family moved to Trent, the home of the mother.

Bruno de Finetti studied applied mathematics in Milan and graduated in 1927 with a dissertation on affine vector spaces from. The work was excellent. He then held research tasks at the newly founded Istituto Centrale di Statistica, Rome and got 1930 the venia legendi. At that time he was also in contact with the Italian form of pragmatism, which greatly influenced him, particularly in his anti-realism. This brought de Finetti to reject the assumption that probabilities are objectively present. Instead, he independently developed by Frank Ramsey, the theory of subjective probability.

1931 was de Finetti in the economy and became the actuary of the Assicurazioni Generali di Trieste. In this context, he also took over teaching activities. In 1939, a call from the University of Trieste, the de Finetti but could not afford episode in Fascist Italy, he was unmarried. In 1946 he was able to take the chair. To this period belong many of his 200 publications, but outside of Italy for a long time remained unheeded.

1951 and 1957 respectively Leonard Jimmie Savage invited him, an influential American statistician, a visiting professor to Chicago one. This left de Finettis conceptions wide notoriety, and named after him sentence (see below) became a mainstay of subjective probability theory.

From 1954 to 1981 de Finetti taught at the University of Rome, where he died highly honored in 1985.

Work

De Finetti suggested the following thought experiment to justify the subjective probability to that conclusion:

You have to set a price for it, if there was life on Mars 10 billion years ago. If this is the case, then you have to pay a dollar, there was no life, so there is no cash flow. The answer to the question of whether there was life, is only revealed in the next day.

If in the allocation of odds ( odds ) is against the rules of probability theory, the other side has a safe way to inflict financial loss to the bookmaker, as de Finetti shows. The rules of probability theory thus also extend to situations of incomplete information, where no random events play a role. For de Finetti probabilities are therefore the product of our lack of information:

" There is no objective probability. "

Commutativity

De Finetti was concerned with commutes in the order (or permutable ) random variables. These are random variables, where the order of events does not affect the overall probability. The assumption of exchangeability is stronger than that the random variables identically distributed, and weaker, than that they are identically distributed and independent.

More

1929 led de Finetti the notion of infinite divisibility of random variables. It is closely connected with that of the Lévy process.

The de Finetti diagrams for simple representation of the proportion of genotypes in a population have been named after him. They are a 2- simplex.

De Finetti rejected the σ - additivity of random variables, because they led to paradoxical consequences of his opinion. Together with Alfréd Rényi he tried to develop an alternative axiomatization of probability theory. However, this axiomatization was hardly popular.

Set of de Finetti

Moreover, he proved in 1931 the rate of de Finetti (English: de Finetti 's theorem or de Finetti 's representation theorem ), which states that all infinitely continuable consequences of interchangeable random variables can be represented as a weighting of an identically and independently distributed random variables - and vice versa.

Suppose an arbitrarily continuable to process, in which there are from tests results and failures, the probability of the number of hits does not depend on their order ( for a given there are consequences from the total possible consequences due to the commutativity ). Then there exists a distribution function such that

Or written as the density of the binomial distribution:

Evidence

De Finetti leads to a heuristic consideration. Suppose first a finite case with experiments in which there are hits and failures. Then each can be considered independently of the others. For a given are all the consequences that this supply of hits, because of the commutativity equally likely. There is an urn model without replacement, and consequently to a hypergeometric distribution. Accordingly, it must be exactly one scaling with and give so that:

When goes to infinity, is the hypergeometric to the binomial distribution over, and the sum becomes the integral. This results in the set.

A formally correct proof can be given about about the problem of moments. The th moment of is equal to the probability of obtaining results from experiments. and the result is thus uniquely determined.

Application

Extensions of the set of random variables with more than two states and processes in which subsequences are interchangeable, and binomial approximation formulas for a finite resumable random variables exist. Meanwhile, it has been shown that finite and negatively correlated random sequences according to de Finetti can be displayed when a signed measure from is selected as the weighting.

These rates establish a relationship between on the one hand the frequency of real events and on the other hand subjective probability assignments, and it is important that the relationship works both ways. This relationship allows to conclude statistically using the set of Bayes. De Finetti and, following him, the followers of a Bayesian probability view see this as a justification of induction.