B. de Finetti. Theory of Probability, volume 1. John Wiley & Sons, Chichester, 1974. English Translation of Teoria delle Probabilit a.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....prices typically decreases as the amount of relevant information increases. In the special case where every gamble X has a fair price , meaning that the supremum acceptable buying price agrees with the infimum acceptable selling price, we obtain the theory of linear previsions of de Finetti [8]. A linear prevision P on a set of gambles K is a map taking K to the set of real numbers R, such that for all m 0 and n 0, and for any X 1 , Xn and Y 1 , Ym in K, sup 2 n X k=1 G(X k ) m X k=1 G(Y k ) 0: A linear prevision( K; P ) is therefore ....

B. de Finetti. Theory of Probability, volume 1. John Wiley & Sons, Chichester, 1974. English Translation of Teoria delle Probabilit a.

....to ( is a nitely additive probability. Conversely, if( K; P ) is a linear prevision, then it is the restriction to K of some linear prevision de ned on L( 4 The linear previsions are the precise probability models, and they are previsions or fair prices in the sense of de Finetti [15]. We denote the set of all linear previsions on L( by P( 9 With an upper prevision( K; P ) we may associate a set M(P ) of dominated linear previsions as follows: M(P ) fP 2 P( 8X 2 K) P (X) P (X) g: Walley [28, Theorem 3.3.3] has shown that( K; P ) avoids sure loss if and only ....

B. de Finetti. Theory of Probability, volume 1. John Wiley & Sons, Chichester, 1974. English Translation of Teoria delle Probabilita.

....set of relevant possible worlds. It is sufficient to work with equivalence classes of possible worlds. Note that these equivalence classes are a generalization of the equivalence classes presented in [41] for conjunctive events. They are based on the concept of conditional events (see especially [12] and [6] 6.2.2 Definitions We start with some preparative definitions. Let L be a classical knowledge base and let F be a probabilistic knowledge base. We first define the operator S F , which increases L by classical knowledge that trivially follows from L [ F . For this purpose, we need ....

B. de Finetti. Theory of Probability. J. Wiley, New York, 1974.

....1 1 Introduction De Finetti s book making principle entails that a gambler should not endorse preferences that can be linearly combined into a sure loss. A surprising implication is that all uncertainties have to be expressed in terms of additive probabilities, possibly subjective (de Finetti [10, 11, 12]) The principle has, since its discovery, served as a justi. cation of Bayesianism. The main restriction of the book making principle is that it requires outcomes to be expressed in utils, in other words, utility must be linear. This requirement is reasonable for small stakes. Linear ....

de Finetti, Bruno (1974), "Theory of Probability." Vol. I. Wiley, New York.

....arising from physical randomness or lack of knowledge should be modelled using mathematical probability. Our differences concern whether probability has some physical existence as in the propensity theories of Popper (1957) whether it is a subjective construct with no objective existence (De Finetti, 1974, 1975), whether it is a property of populations and infinite sequences of repeated trials (Von Mises, 1957) or whatever. Subjectivists use probability in normative modelling to explore the consistency of beliefs and their evolution in the light of data. Some subjectivists effectively deny the ....

....utility (or loss) functions: see, e.g. Box and Taio (1973) Such approaches to inference can be justified simply by following a constructivist approach to the modelling of belief without any need to mention, much less model preference. One can point to the work of Jeffreys (1961) and, possibly, De Finetti (1974,1975) here. Thus for them probability is fundamental to the exclusion of utility. However, their avoidance of the use of utility may be illusory. Presumably they must avoid all aspects of experimental design (Verdinelli, 1992) More importantly, they offer too narrow a view of Bayesian statistics in ....

De Finetti, B. (1975) Theory of Probability. Volume 2. John Wiley, Chichester.

....arising from physical randomness or lack of knowledge should be modelled using mathematical probability. Our differences concern whether probability has some physical existence as in the propensity theories of Popper (1957) whether it is a subjective construct with no objective existence (De Finetti, 1974, 1975) whether it is a property of populations and infinite sequences of repeated trials (Von Mises, 1957) or whatever. Subjectivists use probability in normative modelling to explore the consistency of beliefs and their evolution in the light of data. Some subjectivists effectively deny the ....

....utility (or loss) functions: see, e.g. Box and Taio (1973) Such approaches to inference can be justified simply by following a constructivist approach to the modelling of belief without any need to mention, much less model preference. One can point to the work of Jeffreys (1961) and, possibly, De Finetti (1974,1975) here. Thus for them probability is fundamental to the exclusion of utility. However, their avoidance of the use of utility may be illusory. Presumably they must avoid all aspects of experimental design (Verdinelli, 1992) More importantly, they offer too narrow a view of Bayesian ....

De Finetti, B. (1974) Theory of Probability. Volume 1. John Wiley, Chichester.

....of gambles and the pointwise scalar multiplication of gambles with real numbers. A linear functional P on L( which is positive (X 0 ) P (X) 0) and has unit norm (P (1) 1) is called a linear prevision on L( 3 A linear prevision is a prevision, or fair price, in the sense of de Finetti [21]. Its restriction to ( is called a ( nitely) additive probability on ( 4 Note that P ( X) P (X) X 2 L( 4 which means that as an upper prevision, P is equal to the corresponding lower prevision, i.e. P is self conjugate. The set of linear previsions on L( is denoted by P( 4 If K is a subset ....

B. de Finetti, Theory of Probability, vol. 1, John Wiley & Sons, Chichester, 1974, English Translation of Teoria delle Probabilita.

.... the result of a measurement therefore ultimately depend on the understanding, critical analysis, and integrity of those who contribute to the assignment of its value [8] This appears to me perfectly in line with the lesson of genuine subjectivism, accompanied by the normative rule of coherence[9]. It is instead surprising to see how many Bayesians seek refuge in stereotyped formulae or to see how many still stick to the frequentistic idea that repeated observations are needed in order to evaluate the uncertainty of a measurement. 5 Rough modelling of realistic priors After these ....

B. de Finetti, "Theory of probability", J. Wiley & Sons, 1974.

....They invariably have a methodology for combining evidences and generating a diagnosis from them. A comparative summary of these theories is presented next. Bayesian theory, being the oldest, is the most well developed one and has become the benchmark against which all other theories are compared [16]. There is a well formalized procedure for implementing a diagnostic system based on Bayesian theory and it is based on the following equation: i = 1, m. 1) where n is the number of evidences and m, the number of faults. The variable F i represents the i th fault, and E j , the j th ....

B. de Finetti, "Theory of Probability", John Wiley & Sons, New York, NY. 1974.

....reasoning. What happens as we proceed from one conditional probability P (AjX) to a second one P (AjX; Y ) by conditioning on a further piece of evidence Probabilistic inference or the revision of beliefs in the light of evidence consists of two steps, a logical one and a numerical one (de Finetti 1974; Kleiter, 1991) 1. REMOVE step, the removal of non instantiated possibilities: the truth of the conditioning event eliminates all possible worlds in which this event is false. If a positive test is observed in one or more patients all possible worlds in which this patient or these patients test ....

....stage of induction deals with the remaining probabilities. Given a possibility space together with a probability distribution defined on it, the removal of possibilities leads also to a loss of probability mass. To fit the new situation the probabilities must be re standardized to sum to 1.0. De Finetti (1974) showed that this re standardization if done in a coherent way automatically follows Bayes theorem. The understanding of the REMOVE step is equivalent to understanding which possibilities should be eliminated and which not. Being able to distinguish the relevant from the irrelevant pieces ....

De Finetti, B. (1974). Theory of probability, Vol I. London: Wiley.

....leads to a general theory of probabilistic reasoning, statistical inference and decision making. The following introduction outlines the main concepts of the theory of Walley (1991) 21] which is based on earlier ideas of Keynes (1921) 14] Smith (1961) 17] Good (1962) 12] de Finetti (1974) [8] and Williams (1976) 29] and mentions some of the open problems and directions for further research. Almost all of the ideas presented here are discussed in greater detail in Walley (1991, 1996a) 21, 23] 1. Lower and Upper Previsions A gamble X is a bounded, real valued quantity whose value ....

....investments offered by traders in financial markets are examples of lower and upper previsions. Indeed the theory of coherence is very closely related to the mathematical theory of arbitrage, which has important applications in finance. Upper and lower previsions generalise de Finetti s (1974) [8] concept of prevision; when P (X) P (X) the common value is called a (precise) prevision and denoted by P (X) It is often natural to regard P (X) and P (X) as upper and lower bounds for some ideal price P (X) that is not known precisely. However, there are many applications of the theory in ....

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B. de Finetti. Theory of Probability. Wiley, London, 1974. volumes 1 and 2.

....prices typically decreases as the amount of relevant information increases. In the special case where every gamble X has a fair price , meaning that the supremum acceptable buying price agrees with the infimum acceptable selling price, we obtain the theory of linear previsions of de Finetti [8]. A linear prevision P on a set of gambles K is a map taking K to the set of real numbers R, such that for all m 0 and n 0, and for any X 1 , Xn and Y 1 , Ym in K, sup 2 Omega n X k=1 G(X k ) Gamma m X k=1 G(Y k ) 0: A linear prevision( Omega ; K; P ) is ....

B. de Finetti. Theory of Probability, volume 1. John Wiley & Sons, Chichester, 1974. English Translation of Teoria delle Probabilit a.

....is not all . It only works in situations where the nice scheme of prior and likelihood is applicable. In many circumstances one can assess a subjective probability directly (try asking a carpenter how much he believes the result of his measurement ) Gamma The coherent bet ( a la de Finetti[27]) forces people to be honest and to make the best (i.e. most objective ) assessments of probability. Gamma It is preferable not to mix up probability evaluation with decision problems 24 . what they thought about probability) 22 There are in fact theorists who assume the lower bounds as ....

B. de Finetti, "Theory of probability", J. Wiley & Sons, 1974.

....1 But many other scientists, usually prominent ones, do. And, paradoxically, objective science is, for those who avoid the word belief , nothing but the set of beliefs held by the most influential scientists in whom they believe. enters the game is that of de Finetti s coherent bet [4]. The coherent bet plays the crucial role of neatly separating subjective from arbitrary . In fact, coherence has the normative role of forcing people to be honest and to make the best (i.e. the most objective ) assessments of their degree of belief 2 . Finally comes Bayes rule [9] which ....

B. de Finetti, "Theory of probability", J. Wiley & Sons, 1974.

....first ten tosses. We write p(e 2 je 1 ; to denote the probability of event e 2 given that event e 1 is true and given background knowledge . Many researchers have written down different sets of properties that should be satisfied by degrees of belief (e.g. Cox 1946; Good 1950; Savage 1954; DeFinetti 1970) From each of the lists of properties, these researchers have derived the same rules the rules of probability. Two basic rules, from which other rules may be derived, are the sum rule, which says that for any event e and its complement e, p(ej ) p( ej ) 1 and the product rule, which says ....

....exist corresponding conjugate priors that offer convenient properties for learning probabilities similar to those properties of the Dirichlet. These priors are sometimes referred to collectively as the exponential family. The reader interested in learning about these distributions should read DeGroot (1970, Chapter 9) 5 Learning Probabilities: Known Structure The notion of a random sample generalizes to domains containing more than one variable as well. Given a domain U = fx 1 ; x n g, we can imagine a multivariate physical probability distribution for U . If U contains only discrete ....

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de Finetti, B. (1970). Theory of Probability. Wiley and Sons, New York.

....real numbers. The set of the gambles on Omega is denoted by L( Omega Gamma3 A subset A of Omega is called an event. Its indicator, or characteristic function, A can be interpreted as a 0 Gamma 1 valued gamble, and we shall often identify A with A , a convention which goes back to de Finetti [9]. In such cases it should be clear from the context whether A denotes an event (a set) or a gamble (an indicator) The set of events is denoted by ( Omega Gamma7 An upper prevision P [24] can be formally defined as a real valued function on a class of gambles K L( Omega Gamma9 In order to ....

....a detailed exposition of the theory of imprecise probabilities, we refer to Walley s book [24] A good working knowledge of the material covered there is more or less required for a proper understanding of much of what follows. 3 THE BAYESIAN APPROACH From the so called Bayesian point of view [9], there is a third (internal) rationality criterion that must be satisfied besides avoiding sure loss and coherence: upper and lower prevision should coincide on their common domain 3 (self conjugacy) and therefore be precise. Walley has argued that, whereas avoiding sure loss and coherence are ....

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B. de Finetti. Theory of Probability. John Wiley & Sons, Chichester, 1974. English Translation of Teoria delle Probabilit`a.

.... possible, they are possibilistically independent if and only if 8 : A C 1 6= A C 2 6= A C 3 6= A C 4 6= 8) where the events C 1 def = B C, C 2 def = coB C, C 3 def = B coC and C 4 def = coB coC are called the constituents of B and C (see, for instance, [6]) Let us now assume that the occurrence of B and C is possible but not necessary and therefore uncertain , and look for an interpretation of (8) From our assumption, we easily deduce that 8 : A 6 C 1 A 6 C 2 A 6 C 3 A 6 C 4 : 9) Indeed, assume that for instance A C 3 , ....

....are strictly possible. We may therefore conclude that in this case (8) holds if and only if the constituents C 1 , C 2 , C 3 and C 4 are strictly possible, i.e. are uncertain events. In this case, the uncertain events B and C are in the literature called logically independent (see, for instance, [6] Section 2.7) This logical independence means that additional knowledge about the occurrence of either event B or C can on no account change the existing uncertainty about the occurrence of the other event. To illustrate this, let us assume that (8) holds and that we know that the event B ....

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B. de Finetti, Theory of Probability, John Wiley & Sons, New York, 1974.

.... a nonconcentrated distribution, largely determined by the remaining model parameters, where OE k takes on general values but with a total probability O( ffl) Such a division of probabilities is exactly how various Poisson distributions avoid the Gaussian vise grip of the central limit theorem [6]. It is entirely reasonable that an extra term should appear in the renormalized lattice action the purpose of which is to reduce the probability of large field values. Recall, for classical functions, that the Sobolev inequality [ Z OE 4 (x)d n x] 1 2 C Z [ rOE(x) 2 m 2 OE 2 ....

....stated purposes, and it has the further advantage that it too may be interpreted as a renormalization counterterm for the kinetic energy. The coefficients must be chosen so that, like the case for n = 1, the distribution is a generalized Poisson process, rather than just a compound Poisson process [6]. It may even be possible to determine P 0 to a certain degree based on high temperature series expansions that exist for a general, even, single site field distribution [5] One may try to determine the necessary ffl dependence of the moments of the single site field distribution which ensures ....

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B. de Finetti, Theory of Probability, Vol. 2 (John Wiley & Sons, London, 1975).

....and possibility measures are set mappings. This means that they are defined on specific collections of subsets of a universe. For probability measures these collections are generally oe fields [11, 33] although many subjectivists will defend the definition of probability measures on fields [30]. Possibility measures were originally defined on the power set of a universe, but their domains can very easily be extended 1 towards the more general ample fields [18, 24, 44] An ample field R on a universe X is a collection of subsets of X that is closed under complementation and under ....

B. de Finetti, Theory of Probability (John Wiley and Sons, New York, 1974).

....have been studied intensely. With regards to the first question, many researchers have suggested different sets of properties that should be satisfied by degrees of belief (e.g. Ramsey 1931, Cox 1946, Good Figure 1: The probability wheel: a tool for assessing probabilities. 1950, Savage 1954, DeFinetti 1970). It turns out that each set of properties leads to the same rules: the rules of probability. Although each set of properties is in itself compelling, the fact that different sets all lead to the rules of probability provides a particularly strong argument for using probability to measure beliefs. ....

de Finetti, B. (1970). Theory of Probability. Wiley and Sons, New York.

....(attacks) even though she has complete knowledge and control of these choices. It seems clear that, since they are based upon information which is incomplete in different ways, these two subjective views need not coincide; indeed it would be surprising if they did. It has been argued convincingly [Finetti 1975; Lindley 1985] that such subjective uncertainty should be represented by subjective (Bayesian) probability. The two different viewpoints here will then be represented by different subjective probabilities for the same events. In fact, not only will there be different probabilities associated with ....

B.d. Finetti. Theory of Probability, Chichester, Wiley, 1975.

....seems most appropriate. It should be emphasised that the subjective interpretation of probability is not in any way less scientific than the more usual interpretation based on limiting relative frequency. The present paper is not the place to argue this case; interested readers should consult [Finetti 1975; Lindley 1985] This approach via the subjective interpretation of probability seems particularly appropriate in the case of security. Here we must contend with the possibility of more Towards Operational Measures of Computer Security 14 ....

B. de Finetti. Theory of Probability, Chichester, Wiley, 1975.

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B. de Finetti. Theory of Probability. Wiley, New York, 1974.

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B. de Finetti. Theory of Probability. Wiley, New York, 1974.

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B. de Finetti. Theory of Probability. Wiley, New York, 1974.

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De Finetti, B. (1974). Theory of Probability (2 volumes). Wiley, London.

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B. de Finetti. Theory of Probability. J. Wiley, New York, 1974.

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B. de Finetti. Theory of Probability. J. Wiley, New York, 1974.

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B. de Finetti, Theory of Probability, Wiley, 1974.

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B. de Finetti. Theory of Probability. Wiley, New York, 1974.

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B. de Finetti. Theory of Probability. J. Wiley, New York, 1974.

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B. de Finetti. Theory of Probability, volume 2. Wiley, London, 1975.

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B. de Finetti. Theory of Probability. J. Wiley, New York, 1974.

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B. de Finetti. Theory of Probability. Wiley, New York, 1974.

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B. de Finetti. Theory of Probability. Wiley, New York, 1974.

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B. de Finetti. Theory of Probability. Wiley, New York, 1974.

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