# De Finetti, Bruno

# DE FINETTI, BRUNO

(*b*. Innsbruck, Austria, 13 June 1906;

*d*. Rome, Italy, 20 July 1985), *probability theory, statistics, mathematics*.

De Finetti was a probability theorist who contributed to statistics, mathematics, financial and actuarial science, and economics. He pioneered the subjectivistic interpretation of probability, according to which probabilities describe a person’s propensity to bet, not objective frequencies independent of humans. He also proved a famous representation theorem concerning exchangeable probability assignments, which ground common patterns of inductive inference.

Although he was born in Austria, de Finetti’s parents were Italian. He pursued his university education in Italy and graduated in mathematics from Milan University in 1927. During his academic career he held chairs at Trieste University and the University of Rome. In a long series of publications he promoted the account of subjective probability prominent in decision theory. His views resemble those of Frank Ramsey and Leonard Savage, but he formulated them independently and they have distinctive features. Some probability theorists hold that subjective probabilities are compatible with objective probabilities, but de Finetti denied the existence of objective probabilities and advanced subjective probabilities as a replacement for them. His *Theory of Probability* ([1970] 1974–1975) is a comprehensive presentation of his position on probability.

**Subjective Probability** . De Finetti held that probability exists because of ignorance of events and does not have a foundation in the objective features of events. The illusion of objective probabilities may be explained by properties of subjective probabilities.

According to de Finetti, a person’s probability assignment is defined in terms of betting ratios. These are ratios of stakes that bettors risk. Suppose that one person offers to bet that an event occurs and another person accepts the bet. The person betting that the event occurs stakes *x*, and the person betting that the event does not occur stakes *y*. The betting ratio for the bet is *x*/(*x* + *y*). The probability a person assigns to an event is a betting ratio for the event that the person is willing to use no matter whether he is betting that the event occurs or betting that it does not occur. For example, suppose that a filly Fleetfoot is in a race at Belmont Park today. The probability for a person that Fleetfoot wins depends on the person’s betting ratio for that event. Suppose that the person is willing to pay a maximum of $0.80 for a bet that pays $1.00 if Fleetfoot wins and $0 if she does not win. Then the person adopts a betting ratio of $0.80/($0.80 + $0.20), that is, 0.80, for bets that Fleetfoot wins. Hence the probability of Fleet-foot’s winning is 0.08 for that person.

A set of betting ratios for events is coherent if and only if it does not permit a Dutch book, that is, a system of bets that guarantees a loss. De Finetti showed that coherence among betting ratios requires that they conform to the standard laws of the probability calculus. A coherent set of betting ratios yields a subjective probability function on a set of events. Subjective probabilities obey the laws of probability and so may replace objective probabilities in probability theory.

De Finetti defined conditional probabilities in terms of conditional bets. A bet that Fleetfoot will win if the favorite withdraws is a conditional bet. It is called off if the condition is not met. For coherence, a betting ratio for a conditional bet must equal a ratio of betting ratios for nonconditional bets. The ratio must conform with the usual account of conditional probability, according to which, for events *A* and *B* and a probability assignment *P*, the following equality holds: *P*(*A* given *B*) = *P*(*A* & *B*)/*P*(*B*).

The version of the subjective theory of probability that de Finetti formulated included finite additivity of probabilities but not countable additivity of probabilities. Many probability theorists view countable additivity as an essential component of probability theory. De Finetti excluded it because coherence among betting ratios does not require it. The requirement of coherence yields exactly the laws of probability, he held. As Jan von Plato (1994) observes, de Finetti argued that because coherence requires only finite additivity, countable additivity is optional.

Some probability theorists argue that many people do not have a precise assignment of betting ratios for events and so do not have a probability function on events. To accommodate this observation, de Finetti allowed replacing a single value for an event’s betting ratio with upper and lower bounds for the betting ratio.

De Finetti’s definition of subjective probability met the operational standards of his era. An alternative interpretation of his definition takes it as a method of inferring probabilities from betting ratios instead of as a method of specifying the meaning of subjective probability. A subjective probability may be a rational degree of belief rather than a betting ratio in a coherent system of betting ratios. Rational degrees of belief conform to the laws of probability. Assuming that degrees of belief govern betting behavior, an agent has a reason to hold degrees of belief conforming to those laws. Failure to conform creates the possibility of a Dutch book. This argument for conformity assumes that prevention of Dutch books is an optimal strategy. However, prevention of Dutch books may have costs that make it sub-optimal. For example, preventing Dutch books also prevents systems of bets that guarantee gains. Whether prevention of Dutch books is optimal depends on circumstances, as Alan Hájek (2005) observes. Fortunately, besides the Dutch book argument there are nonpragmatic and purely epistemic arguments for holding degrees of belief that conform to the probability laws, at least in the case of ideal agents. These arguments show that conformity is a cognitive goal for reasonable people.

As von Plato explains, de Finetti took two approaches to quantitative probability. The first, already described, defined it in terms of betting ratios. The second, intended as a complementary approach, defined it using qualitative, or comparative, probability. For the second method, de Finetti formulated axioms that govern comparisons of probability and showed that if a person satisfies those axioms, there is a unique probability function that represents the person’s comparisons of probability.

**Exchangeability** . De Finetti contributed to probability theory major results concerning exchangeability and used them to explain inductive inference. A probability assignment for sequences of trials is exchangeable if and only if it assigns the same probability to all sequences of the same length and number of successes. Exchangeability implies that for all positive integers *n*, all permutations of the results of a sequence of length *n* have the same probability. Hence a sequence with exactly two successes followed by a failure has the same probability as a sequence with exactly two successes separated by a failure. If a probability assignment is exchangeable, it displays a type of symmetry. Also, the success rate in a sequence of trials is a sufficient statistic, that is, it captures all information about the sequence relevant to inferences about success rates in future sequences. For example, given an exchangeable probability assignment for sequences of trials, the probability of a result on the next trial depends only on the number of previous trials and the number of successes in those trials. If probabilities of sequences of a biased coin’s tosses are exchangeable, then after observing a sequence of tosses, one needs only the number of heads and tails in the sequence to assess the probability that the next toss of the coin will yield heads.

Theorists also attribute exchangeability directly to sequences of trials. For example, a set of sequences of coin tosses has the property of exchangeability if the same probability attaches to obtaining a certain number of heads in all sequences of the same length. In general, a sequence of random variables (which de Finetti preferred to call random quantities) has the property of exchangeability if the probability assignment governing it is exchangeable.

According to Sandy L. Zabell (2005, pp. 9, 56), William Ernest Johnson introduced exchangeability. However, de Finetti made exchangeability prominent by proving a major representation theorem involving it. Zabell’s notation (p. 3) is being used here to present the theorem.

An indicator random-variable uses numbers to indicate whether an event occurs, for example, whether a coin toss yields heads. The number 1 may stand for heads and the number 0 may stand for tails. Let *X*_{1}, *X*_{2}, *X*_{3} … be an infinite sequence of indicator random-variables having the value 0 or 1. The sequence is exchangeable if all finite subsequences of the same length with the same number of ones have the same probability.

De Finetti showed that if the infinite sequence *X*_{1}, *X*_{2}, … X_{n,}, … is exchangeable, then the relative frequency of successes in *n* trials, that is the number of successes divided by *n*, will with probability 1 have a limit as *n* goes to infinity. Also, a mixture of binomial probabilities having that random limit *Z* as success parameter represents the probability distribution of the sequence. In other words, given any exchangeable probability assignment for finite sequences, there is a unique probability distribution over a mixture of binomial probabilities that gives the probability of *r* successes in *n* trials. This theorem is called a representation theorem because the probability distribution represents the consequences of an exchangeable probability assignment. As von Plato remarks, de Finetti held that the representation theorem shows how to replace unknown objective probabilities of independent events with subjective probabilities.

As a corollary of his representation theorem, de Finetti showed that in almost every case, after observing a sufficiently long initial segment of the sequence *X*_{1}, *X*_{2}, … X_{n,}, …, the posterior distribution of the limit *Z* will be highly peaked about the observed relative frequency, and future trials will be expected to occur with a relative frequency very close to the observed relative frequency. As Zabell explains, de Finetti used this mathematical result as a justification of inductive reasoning. Suppose that a person has coherent attitudes toward events so that her attitudes yield a probability function for the events. Also, suppose that with respect to the probability function, future events form an exchangeable sequence. Then if an event occurs with a certain relative frequency in a long sequence of trials, the person will assign a high probability to that event’s occurring with approximately the same relative frequency in remaining future trials. In that sense, the person will expect the future to resemble the past.

Given an exchangeable probability assignment, the probability of success on a trial given a sequence of previous trials depends solely on the number of successes observed and does not depend on their order in the sequence. Exchangeability thus rules out sensible forms of inductive inference that attend to the order of successes. An exchangeable probability assignment is warranted only when order has no significance, as with a sequence of coin tosses. To treat other cases, de Finetti introduced a generalization of exchangeability he called partial exchangeability. Persi Diaconis and David Freedman (1980) evaluate partial exchangeability. Other generalizations of exchangeability taking account of the effects of order are topics of current research in statistics. As Zabell notes, to each type of exchangeability corresponds a judgment concerning a type of symmetry, the sufficiency of a statistic, a representation theorem, and a corresponding method of inductive inference.

De Finetti had multiple talents, and his contributions outside probability theory are numerous. Some of his early work was in genetics. It introduced de Finetti diagrams, which are still used for graphing genotype frequencies. Also, de Finetti formulated a mean-variance theory of portfolio selection. To evaluate an investment, it combines an assessment of the investment’s expected return with an assessment of its risk. Mean-variance portfolio selection seeks the best overall balance of expected return and risk. De Finetti’s theory is among the earliest theories of this type.

De Finetti received many honors, including honorary membership in the Royal Statistical Society. In 2006 the University of Rome organized an international symposium called the Bruno de Finetti Centenary Conference. Speakers demonstrated the continuing influence of de Finetti’s ideas in a variety of fields.

## BIBLIOGRAPHY

*For listings of de Finetti’s works, consult the Bruno de Finetti Web site (http://www.brunodefinetti.it/index_en.htm), the bibliography in von Plato (1994), and the Bruno de Finetti Collection at the University of Pittsburgh*.

### WORKS BY DE FINETTI

“Probabilism: A Critical Essay on the Theory of Probability and on the Value of Science” [1931]. Translated by Maria Concetta di Maio, Maria Carla Galavotti, and Richard Jeffrey. *Erkenntnis* 31 (1989): 169–223.

“Foresight: Its Logical Laws, Its Subjective Sources” [1937]. Translated by Henry Kyburg. In *Studies in Subjective Probability*, edited by Henry Kyburg and Howard Smokler, 93–158. New York: Wiley, 1964.

“On the Condition of Partial Exchangeability” [1938]. Translated by Paul Benacerraf and Richard Jeffrey. In *Studies in Inductive Logic and Probability*, vol. 2, edited by Richard Jeffrey, 193–205. Berkeley: University of California Press, 1980.

*Theory of Probability* [1970]. 2 vols. Translated by Antonio Machí and Adrian Smith. New York: Wiley, 1974–1975.

*Probability, Induction, and Statistics: The Art of Guessing*. New York: Wiley, 1972.

### OTHER SOURCES

Diaconis, Persi, and David Freedman. “De Finetti’s Generalization of Exchangeability.” In *Studies in Inductive Logic and Probability*, vol. 2, edited by Richard Jeffrey, 233–249. Berkeley: University of California Press, 1980.

Galavotti, Maria Carla. “The Notion of Subjective Probability in the Work of Ramsey and de Finetti.” *Theoria* 57 (1991): 239–259.

———, and Richard Jeffrey, eds. “Bruno de Finetti’s Philosophy of Probability.” Double issue. *Erkenntnis* 31, nos. 2, 3 (1989). The essays in this collection treat de Finetti’s interpretation of probability and his mathematical results concerning exchangeability.

Hájek, Alan. “Scotching Dutch Books?” *Philosophical Perspectives* 19 (2005): 139–151.

Jeffrey, Richard. “Reading *Probabilismo*.” *Erkenntnis* 31 (1989): 225–237. A discussion of de Finetti ([1931] 1989).

———. “Conditioning, Kinematics, and Exchangeability.” In his *Probability and the Art of Judgment*, 117–153. Cambridge, U.K.: Cambridge University Press, 1992.

Milne, Peter. “Bruno de Finetti and the Logic of Conditional Events.” *British Journal for the Philosophy of Science* 48, no. 2 (1997): 195–232.

von Plato, Jan. *Creating Modern Probability: Its Mathematics, Physics and Philosophy in Historical Perspective*. Cambridge, U.K.: Cambridge University Press, 1994. Chapter 8 treats de Finetti’s theory of subjective probability and his representation theorem involving exchangeability.

Zabell, Sandy L. *Symmetry and Its Discontents: Essays on the History of Inductive Philosophy*. Cambridge, U.K.: Cambridge University Press, 2005. Chapters 1 and 2 treat exchangeability, de Finetti’s representation theorem involving it, and the theorem’s implications for inductive reasoning.

*Paul Weirich*

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