Bruno de Finetti

Bruno de Finetti (born June 13, 1906 in Innsbruck , † July 20, 1985 in Rome ) was an Italian mathematician . His most important results can be found in statistics and probability theory. In particular, he developed the subjective concept of probability .

Life

De Finetti was born as Bruno Johannes Leonhard Maria von Finetti. His grandfather, Giovanno Ritter von Finetti, was a building contractor from Trieste , which at that time belonged to Austria, and was involved in the construction of the Arlbergbahn . That's why the family moved to Innsbruck. De Finetti's father Gualtiero took over the company until he was appointed building director in Trieste in 1910. Shortly afterwards the father died and the family moved to Trento , the mother's home.

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

In 1931 de Finetti went into business and was appointed actuary of the Assicurazioni Generali di Trieste . In this context, he also took on teaching activities. In 1939 a call from the University of Trieste followed , which de Finetti could not obey in Fascist Italy because he was unmarried. He was only able to accept the chair in 1946. Many of his 200 publications fall during this period, but they went unnoticed outside of Italy for a long time.

In 1951 and 1957, Leonard Jimmie Savage , an influential American statistician, invited him to Chicago for a visiting professorship . As a result, de Finetti's views became widely known, and the sentence named after him ( see below ) became a mainstay of the subjective theory of probability.

From 1954 to 1981 de Finetti taught at the University of Rome , where he died in 1985 with great honor.

plant

De Finetti proposed the following thought experiment to justify the subjective conception of probability :

You have to set a price for whether there was life on Mars 10 billion years ago. If this is the case, then one has to pay a dollar; if there was no life, there is no flow of money. The answer to the question of whether there was life will not be revealed until the next day.

If the rules of probability are broken when assigning the odds , the other side has a sure way to inflict financial loss on the bookmaker , as de Finetti shows. The rules of the calculus of probability 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 insufficient information:

"There is no objective probability."

- Bruno de Finetti

Interchangeability

De Finetti dealt with random variables that could be exchanged in sequence , so-called exchangeable families of random variables . These are random variables for which the sequence of events has no influence on the overall probability. The assumption of interchangeability is stronger than that the random variables are identically distributed, and weaker than that they are identically distributed and independent.

additional

In 1929 de Finetti introduced the concept of the infinite divisibility of random variables. It is closely linked to that of the Lévy trial .

The De Finetti diagrams for the simple representation of the proportion of genotypes in a population were named after him. You are a 2 simplex .

De Finetti rejected the σ-additivity of random variables because, in his view, it led to paradoxical consequences. Together with Alfréd Rényi , he tried to develop an alternative axiomatization of the calculus of probability. However, this axiomatization hardly met with approval.

De Finetti's theorem

He also showed in 1931 the rate of de Finetti (also representation theorem of de Finetti , English: de Finetti's theorem or de Finetti's representation theorem ), which states that all continuable to infinity consequences of interchangeable random variables are presented as weighting an identically and independently distributed random variables can - and vice versa.

Let us assume a process that can be continued at will in which attempts result in hits and failures, the probability of a number of hits not depending on their sequence (for a given one there are consequences out of the total possible consequences due to the interchangeability ). Then there is exactly one distribution function such that ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle n-k}$ ${\displaystyle k}$ ${\displaystyle {\tbinom {n}{k}}}$ ${\displaystyle 2^{n}}$ ${\displaystyle W(p)}$

${\displaystyle P(s_{n}=k)={n \choose k}\int _{0}^{1}p^{k}(1-p)^{n-k}\mathrm {d} W(p)}$,

or with written as the density of the binomial distribution : ${\displaystyle bin(k|p,n)}$

${\displaystyle P(s_{n}=k)=\int _{0}^{1}bin(k|p,n)\mathrm {d} W(p)}$.

proof

De Finetti puts forward a heuristic consideration. Let us first take a finite case with trials, where there are hits and failures. Then each can be viewed independently of the others. For a given one , all of the consequences that it produces in terms of hits are equally probable because of the interchangeability. It is an urn model with no replacement and therefore a hypergeometric distribution . Accordingly, there must be exactly one weighting with and so that: ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle n-k}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle w}$ ${\displaystyle 0\leq w(k)\leq 1}$ ${\displaystyle \textstyle \sum _{k=0}^{n}w(k)=1}$

${\displaystyle P(s_{n}=k)=\sum _{k=0}^{n}hyp(k|n,...)w(k)}$

When it approaches infinity, the hypergeometric distribution goes over to the binomial distribution and the sum becomes an integral. This gives the sentence. ${\displaystyle n}$

A formally correct proof can be given for example using the moment problem . The -th moment of equals the probability of getting hits from attempts . and the consequences are clearly determined by it. ${\displaystyle n}$ ${\displaystyle W(p)}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle W(p)}$

application

Extensions of the set to random variables with more than two states and processes in which partial sequences are interchangeable, as well as binomial approximation formulas for random variables that can only be continued finitely exist. Meanwhile, it was shown that also finite and negatively correlated random sequences according to de Finetti can be represented as a weighting if a signed measure of is selected. ${\displaystyle [-1,1]}$

These sentences establish a connection between, on the one hand, the frequency of real events and, on the other hand, subjective assignments of probability, whereby it is important that the connection works in both directions. This relationship allows statistical conclusions to be drawn using Bayes' theorem . De Finetti and, following him, the supporters of a Bayesian concept of probability see this as a justification for induction .

Web links

• Website dedicated to de Finetti's memory
• Generate De Finetti diagrams online for free
• Literature by and about Bruno de Finetti in the catalog of the German National Library
• John J. O'Connor, Edmund F. Robertson :  Bruno de Finetti. In: MacTutor History of Mathematics archive .
• Probabilità e induione. Bologna, 1993.

Individual evidence

1. ↑ De Finetti's biography of his cousin Dr. Hans Hochenegg (PDF; 1.5 MB).
2. ↑ Bruno de Finetti: Probability Theory. Preface. Oldenbourg (Munich) 1981.
3. ↑ Ibid., P. 618, footnote 18:

   “Those who follow a series of syllogisms or small transitions [...] can be led to reluctantly acknowledge a truth without seeing why . But to see exactly why is, in my opinion, the essential thing [...] "

4. ↑ Ibid, p. 618.
5. ↑ Richard Jeffrey: Subjective Probability (The Real Thing). Cambridge University Press 2004 (PDF; 605 kB), p. 87.
6. ↑ Jay Kerns, Gábor Székely: De Finetti's Theorem for Finite Exchangeable Sequences. 2005.
7. ^ Jose M. Bernardo: The Concept of Exchangeability and its Applications. 1996 (PDF; 87 kB)
8. ↑ Kerns, Székely, op. a. Oh, fourth chapter.
9. ^ De Finetti, op. a. Oh, chapter 11.