Bruno de Finetti (1906–1985) was one of the most original and provocative thinkers in the history of probability. An Italian mathematician and actuary, he developed a rigorously subjectivist theory of probability in which probabilities express nothing more—and nothing less—than an individual's coherent degrees of belief. His representation theorem for exchangeable sequences provided the mathematical bridge between subjective belief and statistical modeling, and his philosophical boldness helped galvanize the twentieth-century Bayesian revival.
Early Life and Education
De Finetti was born in Innsbruck, Austria (then part of the Austro-Hungarian Empire), to an Italian family. He studied mathematics at the Polytechnic University of Milan, graduating in 1927. Even as a student, he was drawn to probability theory and its philosophical foundations, publishing his first significant paper on exchangeability at the age of twenty-three.
“Probability Does Not Exist”
De Finetti opened his monumental 1970 treatise Teoria delle Probabilità with the provocative declaration: “PROBABILITY DOES NOT EXIST.” Written in capital letters, this statement encapsulated his philosophical position. He meant that probability is not an objective property of events or physical systems; rather, it is a measure of an individual's uncertainty, constrained only by the requirement of coherence (consistency with the axioms of probability). This position placed him firmly in the subjectivist tradition alongside Ramsey and Savage, though de Finetti arrived at his views largely independently.
“PROBABILITY DOES NOT EXIST. The abandonment of superstitious beliefs about the existence of Phlogiston, the Cosmic Ether, Absolute Space and Time, ... or Fairies and Witches, was an essential step along the road to scientific thinking. Probability, too, if regarded as something endowed with some kind of objective existence, is no less a misleading misconception.”— Bruno de Finetti, Theory of Probability (1970)
The Representation Theorem
De Finetti's most celebrated mathematical result is his representation theorem for exchangeable sequences, first published in 1931 and refined in 1937. The theorem states that any infinite sequence of exchangeable random variables (variables whose joint distribution is invariant under permutations) can be represented as a mixture of independent and identically distributed sequences. Formally, for an exchangeable sequence, the joint probability can be written as an integral over a mixing measure, which in Bayesian terms corresponds to a prior distribution over the unknown parameter.
This result was revolutionary because it showed that the Bayesian framework—with its prior distributions and likelihood functions—arises naturally from the simple symmetry assumption of exchangeability, without any need to postulate objective probabilities or “true” parameter values. The parameter θ is, in de Finetti's view, merely a mathematical convenience, not a real quantity.
Exchangeability is weaker than independence: it requires only that the joint distribution be invariant under permutation of the observations, not that the observations be unrelated. De Finetti showed that exchangeability implies conditional independence given the mixing parameter—providing a subjectivist justification for the standard statistical assumption of i.i.d. sampling from an unknown distribution.
Career and Influence
De Finetti worked as an actuary at the Assicurazioni Generali insurance company in Trieste from 1927 to 1931, then held academic positions in Trieste, Padua, and eventually Rome. His actuarial experience deeply informed his probabilistic thinking: insurance is a domain where subjective assessment of risk is not just philosophically interesting but practically essential. He was a prolific writer, producing over three hundred papers and several books, and was active in debates about the foundations of probability throughout his life.
Born on 13 June in Innsbruck, Austria.
Graduated from the Polytechnic University of Milan; began work at Assicurazioni Generali.
Published first version of the representation theorem for exchangeable events.
Published the definitive version: “La prévision: ses lois logiques, ses sources subjectives.”
Appointed professor at the University of Rome.
Published Teoria delle Probabilità, his comprehensive treatise on subjective probability.
Died in Rome on 20 July.