Decision Making for Inconsistent Expert Judgments Using Negative Probabilities
aa r X i v : . [ s t a t . O T ] S e p Decision Making for Inconsistent ExpertJudgments Using Negative Probabilities
J. Acacio de Barros
Liberal Studies Program, San Francisco State University, 1600 Holloway Ave., SanFrancisco, CA 94132
Abstract.
In this paper we provide a simple random-variable exam-ple of inconsistent information, and analyze it using three different ap-proaches: Bayesian, quantum-like, and negative probabilities. We thenshow that, at least for this particular example, both the Bayesian andthe quantum-like approaches have less normative power than the nega-tive probabilities one.
In recent years the quantum-mechanical formalism (mainly from non-relativisticquantum mechanics) has been used to model economic and decision-making pro-cesses (see [1,2] and references therein). The success of such models may originatefrom several related issues. First, the quantum formalism leads to a propositionalstructure that does not conform to classical logic [3]. Second, the probabilitiesof quantum observables do not satisfy Kolmogorov’s axioms [4]. Third, quantummechanics describes experimental outcomes that are highly contextual [5,6,7,8,9].Such issues are connected because the logic of quantum mechanics, representedby a quantum lattice structure [3], leads to upper probability distributions andthus to non-Kolmogorovian measures [10,11,12], while contextuality leads thenonexistence of a joint probability distribution [13,14].Both from a foundational and from a practical point of view, it is importantto ask which aspects of quantum mechanics are actually needed for social-sciencemodels. For instance, the Hilbert space formalism leads to non-standard logicand probabilities, but the converse is not true: one cannot derive the Hilbertspace formalism solely from weaker axioms of probabilities or from quantum lat-tices. Furthermore, the quantum formalism yields non-trivial results such as theimpossibility of superluminal signaling with entangled states [15]. These typesof results are not necessary for a theory of social phenomena [16], and we shouldask what are the minimalistic mathematical structures suggested by quantummechanics that reproduce the relevant features of quantum-like behavior.In a previous article, we used reasonable neurophysiological assumptions tocreated a neural-oscillator model of behavioral Stimulus-Response theory [17].We then showed how to use such model to reproduce quantum-like behavior[18]. Finally, in a subsequent article, we remarked that the same neural-oscillatormodel could be used to represent a set of observables that could not correspond . INCONSISTENT INFORMATION to quantum mechanical observables [19], in a sense that we later on formalize inSection 3. These results suggest that one of the main quantum features relevantto social modeling is contextuality, represented by a non-Kolmogorovian proba-bility measure, and that imposing a quantum formalism may be too restrictive.This non-Kolmogorovian characteristic would come when two contexts providingincompatible information about observable quantities were present.Here we focus on the incompatibility of contexts as the source of a violation ofstandard probability theory. We then ask the following question: what formalismsare normative with respect to such incompatibility? This question comes fromthe fact that, in its origin, probability was devised as a normative theory, and notdescriptive. For instance, Richard Jeffrey [20] explains that “the term ’probable’(Latin probable ) meant approvable , and was applied in that sense, univocally, toopinion and to action. A probable action or opinion was one such as sensiblepeople would undertake or hold, in the circumstances.” Thus, it should comeas no surprise that humans actually violate the rules of probability, as shown inmany psychology experiments. However, if a person is to be considered “rational,”according to Boole, he/she should follow the rules of probability theory.Since inconsistent information, as above mentioned, violates the theory ofprobability, how do we provide a normative theory of rational decision-making?There are many approaches, such as Bayesian models or the Dempster-Shaffertheory, but here we focus on two non-standard ones: quantum-like and negativeprobability models. We start first by presenting a simple case where expertjudgments lead to inconsistencies. Then, we approach this problem first with astandard Bayesian probabilistic method, followed by a quantum model. Finally,we use negative probability distributions as a third alternative. We then comparethe different outcomes of each approach, and show that the use of negativeprobabilities seems to provide the most normative power among the three. Weend this paper with some comments.
As mentioned, the use of the quantum formalism in the social sciences originatesfrom the observation that Kolmogorov’s axioms are violated in many situations[1,2]. Such violations in decision-making seem to indicate a departure from arational view, and in particular to though-processes that may involve irrationalor contradictory reasoning, as is the case in non-monotonic reasoning. Thus,when dealing with quantum-like social phenomena, we are frequently dealingwith some type of inconsistent information, usually arrived at as the end resultof some non-classical (or incorrect, to some) reasoning. In this section we examinethe case where inconsistency is present from the beginning.Though in everyday life inconsistent information abounds, standard classicallogic has difficulties dealing with it. For instance, it is a well know fact thatif we have a contradiction, i.e. A & ( ¬ A ) , then the logic becomes trivial, in thesense that any formula in such logic is a theorem. To deal with such difficulty,logicians have proposed modified logical systems (e.g. paraconsistent logics [21]).2 . INCONSISTENT INFORMATION Here, we will discuss how to deal with inconsistencies not from a logical pointof view, but instead from a probabilistic one.Inconsistencies of expert judgments are often represented in the probabilityliterature by measures corresponding to the experts’ subjective beliefs [22]. It isfrequently argued that this subjective nature is necessary, as each expert makesstatements about outcomes that are, in principle, available to all experts, anddisagreements come not from sampling a certain probability space, but frompersonal beliefs. For example, let us assume that two experts, Alice and Bob,are examining whether to recommend the purchase of stocks in company X ,and each gives different recommendations. Such differences do not emerge froman objective data (i.e. the actual future prices of X ), but from each expert’sinterpretations of current market conditions and of company X . In some casesthe inconsistencies are evident, as when, say, Alice Alice recommends buy, andBob recommends sell; in this case the decision maker would have to reconcilethe discrepancies.The above example provides a simple case. A more subtle one is when theexperts have inconsistent beliefs that seem to be consistent. For example, eachexpert, with a limited access to information, may form, based on different con-texts, locally consistent beliefs without directly contradicting other experts. Butwhen we take the totality of the information provided by all of them and tryto arrive at possible inferences, we reach contradictions. Here we want to createa simple random-variable model that incorporates expert judgments that arelocally consistent but globally inconsistent. This model, inspired by quantumentanglement, will be used to show the main features of negative probabilitiesas applied to decision making.Let us start with three ± -valued random variables, X , Y , and Z , withzero expectation. If such random variables have correlations that are too strongthen there is no joint probability distribution [13]. To see this, imagine theextreme case where the correlations between the random variables are E ( XY ) = E ( YZ ) = E ( XZ ) = − . Imagine that in a given trial we draw X = 1 . From E ( XY ) = − it follows that Y = − , and from E ( YZ ) = − that Z = 1 . Butthis is in contradiction with E ( XZ ) = − , which requires Z = − . Of course,the problem is not that there is a mathematical inconsistency, but that it is notpossible to find a probabilistic sample space for which the variables X , Y , and Z have such strong correlations. Another way to think about this is that the the X measured together with Y is not the same one as the X measured with Z :values of X depend on its context.The above example posits a deterministic relationship between all randomvariables, but the inconsistencies persist even when weaker correlations exist. Infact, Suppes and Zanotti [13] proved that a joint probability distribution for X , Y , and Z exists if and only if − ≤ E ( XY ) + E ( YZ ) + E ( XZ ) ≤ { E ( XY ) , E ( YZ ) , E ( XZ ) } . (1)The above case violates inequality (1). 3 . QUANTUM APPROACH Let’s us now consider the example we want to analyze in detail. Imagine X , Y , and Z as corresponding to future outcomes in a company’s stocks. Forinstance, X = 1 corresponds to an increase of the stock value of company X inthe following day, while X = − a decrease, and so on. Three experts, Alice ( A ),Bob ( B ), and Carlos ( C ), have the following beliefs about those stocks. Aliceis an expert on companies X and Y , but knows little or nothing about Z , soshe only tells us what we don’t know: her expected correlation E A ( XY ) . Bob(Carlos), on the other hand, is only an expert in companies X and Z ( Y and Z ), and he too only tells us about their correlations. Let us take the case where E A ( XY ) = − , (2) E B ( XZ ) = − , (3) E C ( YZ ) = 0 , (4)where the subscripts refer to each experts. For such case, the sum of the correla-tions is − , and according to (1) no joint probability distribution exists. Sincethere is no joint, how can a rational decision-maker decide what to do whenfaced with the question of how to bet in the market? In particular, how can sheget information about the joint probability, and in particular the unknown triplemoment E ( XYZ ) ? In the next sections we will show how we can try to answerthese questions using three possible approaches: quantum, Bayesian, and signedprobabilities. We start with a comment about the quantum-like nature of correlations (2)-(4).The random variables X , Y , and Z with correlations (2)-(4) cannot be repre-sented by a quantum state in a Hilbert space for the observables correspondingto X , Y , and Z . This claim can be expressed in the form of a simple proposition. Proposition 1.
Let ˆ X , ˆ Y , and ˆ Z be three observables in a Hilbert space H with eigenvalues ± , let them pairwise commute, and let the ± -valued randomvariable X , Y , and Z represent the outcomes of possible experiments performedon a quantum system | ψ i ∈ H . Then, there exists a joint probability distributionconsistent with all the possible outcomes of X , Y , and Z .Proof. Because ˆ X , ˆ Y , and ˆ Z are observables and they pairwise commute, itfollows that their combinations, ˆ X ˆ Y , ˆ Y ˆ Z , ˆ X ˆ Z , and ˆ X ˆ Y ˆ Z are also observables,and they commute with each other. For instance, (cid:16) ˆ X ˆ Y ˆ Z (cid:17) † = ˆ Z † ˆ Y † ˆ X † = ˆ X ˆ Y ˆ Z. Furthermore, [ ˆ X ˆ Y ˆ Z, ˆ X ] = [ ˆ X ˆ Y ˆ Z, ˆ Y ] = · · · = [ ˆ X ˆ Y ˆ Z, ˆ X ˆ Z ] = 0 . . BAYESIAN APPROACH Therefore, quantum mechanics implies that all three observables ˆ X , ˆ Y , and ˆ Z can be simultaneously measured. Since this is true, for the same state | ψ i wecan create a full data table with all three values of X , Y , and Z (i.e., no missingvalues), which implies the existence of a joint.So, how would a quantum-like model of correlations (2)–(4) be like? The aboveresult depends on the use of the same quantum state | ψ i throughout the manyruns of the experiment, and to circumvent it we would need to use differentstates for the system. In other words, if we want to use a quantum formalismto describe the correlations (2)-(4), a | ψ i would have to be selected for eachrun such that a different state would be used when we measure ˆ X ˆ Y , e.g. | ψ i xy ,than when we measure ˆ X ˆ Z , e.g. | ψ i xz . Then, the quantum description could beaccomplished by the state | ψ i = c A | A i ⊗ | ψ i xy + c B | B i ⊗ | ψ i xz + c C | C i ⊗ | ψ i yz . This state would model the correlations the following way. When Alice makesher choice, she uses a projector into her “state of knowledge” ˆ P A = | A ih A | , andgets the correlation E A ( XY ) , and similarly for Bob and Carlos.In the above example, all correlations and expectations are given, and theonly unknown is the triple moment E ( XYZ ) . Furthermore, since we do nothave a joint probability distribution, we cannot compute the range of values forsuch moment based on the expert’s beliefs. But the question still remains as towhat would be our best bet given what we know, i.e., what is our best guess for E ( XYZ ) . The quantum mechanical approach does not address this question, asit is not clear how to get it from the formalism given that any superposition ofthe states preferred by Alice, Bob, and Carlos are acceptable (i.e., we can chooseany values of c A , c B , and c C ). Here we focus again on the unknown triple moment. As we mentioned before,there are many different ways to approach this problem, such as paraconsis-tent logics, consensus reaching, or information revision to restore consistency.Common to all those approaches is the complexity of how to resolve the incon-sistencies, often with the aid of ad hoc assumptions [22]. Here we show how aBayesian approach would deal with the issue [23,24].In the Bayesian approach, a decision maker, Deanna ( D ), needs to accesswhat is the joint probability distribution from a set of inconsistent expecta-tions. To set the notation, let us first look at the case when there is only oneexpert. Let P A ( x ) = P A ( X = x | δ A ) be the probability assigned to event x byAlice conditioned on Alice’s knowledge δ A , and let P D ( x ) = P D ( X = x | δ D ) beDeanna’s prior distribution, also conditioned on her knowledge δ D . Furthermore,let P A = P A ( x ) be a continuous random variable, P A ∈ [0 , , such that its out-come is P A ( x ) . The idea behind P A is that consulting an expert is similar toconducting an experiment where we sample the experts opinion by observing a5 . BAYESIAN APPROACH distribution function, and therefore we can talk about the probability that anexpert will give an answer for a specific sample point. Then, for this case, Bayes’stheorem can be written as P ′ D ( x | P A = P A ( x )) = P D ( P A = P A ( x )) P D ( x ) P D ( P A = P A ( x )) , where P ′ D ( x | P A = P A ( x )) is Deanna’s posterior distribution revised to take intoaccount the expert’s opinion. As is the case with Bayes’s theorem, the difficultylies on determining the likelihood function P D ( P A ) , as well as the prior. Thislikelihood function is, in a certain sense, Deanna’s model of Alice, as it is whatDeanna believes are the likelihoods of each of Alice’s beliefs. In other words, sheshould have a model of the experts. Such model of experts is akin to giving eachexpert a certain measure of credibility, since an expert whose model doesn’t fitDeanna’s would be assigned lower probability than an expert whose model fits.The extension for our case of three experts and three random variables iscumbersome but straightforward. For Alice, Bob, and Carlos, Deanna needs tohave a model for each one of them, based on her prior knowledge about X , Y ,and Z , as well as Alice, Bob, and Carlos. Following Morris [23], we construct aset E consisting of our three experts joint priors: E = { P A ( x, y ) , P B ( y, z ) , P C ( x, z ) } . Deanna’s is now faced with the problem of determining the posterior P ′ D ( x | E ) , using Bayes’s theorem, given her new knowledge of the expert’s priors.In a Bayesian approach, the decision maker should start with a prior beliefon the stocks of X , Y , and Z , based on her knowledge. There is no recipe forchoosing a prior, but let us start with the simple case where Deanna’s lack ofknowledge about X , Y , and Z means she starts with the initial belief that allcombinations of values for X , Y , and Z are equiprobable. Let us use the followingnotation for the probabilities of each atom: p xyz = P ( X = +1 , Y = +1 , Z = +1) , p xyz = P ( X = +1 , Y = +1 , Z = − , p xyz = P ( X = − , Y = +1 , Z = − , andso on. Then Deanna’s prior probabilities for the atoms are p Dxyz = p Dxyz = · · · = p Dxyz = 116 , where the superscript D refers to Deanna.When reasoning about the likelihood function, Deanna asks what would bethe probable distribution of responses of Alice if somehow she (Deanna) couldsee the future (say, by consulting an Oracle) and find out that E ( XY ) = − .For such case, it would be reasonable for Alice to think it more probable tohave, say, xy than xy , since she was consulted as an expert. So, in terms of thecorrelation ǫ A , Deanna could assign the following likelihood function: P D ( ǫ A | xy ) = P D ( ǫ A | xy ) = 14 (1 − ǫ A ) , (5) P D ( ǫ A | xy ) = P D ( ǫ A | xy ) = 1 −
14 (1 − ǫ A ) , (6)6 . BAYESIAN APPROACH where the minus sign represents the negative, i.e. p Axy · = p xy · = (1 + ǫ A ) and p xy · = p xy · = (1 − ǫ A ) . Notice that the choice of likelihood function is arbi-trary.Deanna’s posterior, once she knows that Alice thought the correlation to bezero (cf. (2)), constitutes, as we mentioned above, an experiment. To illustratethe computation, we find its value below, from Alice’s expectation E A ( XY ) = − . From Bayes’s theorem p D | Axyz = k (cid:20) −
14 (1 − ǫ A ) (cid:21)
18= 14 (cid:20) −
14 (1 − ǫ A ) (cid:21) = 316 , where the normalization constant k is given by k − = (cid:20) −
14 (1 − ǫ A ) (cid:21)
18 + (cid:20)
14 (1 − ǫ A ) (cid:21)
18 + (cid:20)
14 (1 − ǫ A ) (cid:21) (cid:20) −
14 (1 − ǫ A ) (cid:21)
18 + (cid:20)
14 (1 − ǫ A ) (cid:21)
18 + (cid:20)
14 (1 − ǫ A ) (cid:21) (cid:20) −
14 (1 − ǫ A ) (cid:21)
18 + (cid:20) −
14 (1 − ǫ A ) (cid:21) , and we use the notation p D | A to explicitly indicate that this is Deanna’s posteriorprobability informed by Alice’s expectation. Similarly, we have p D | Axyz = p D | Axyz = p D | Axyz = p D | Axyz = 116 , and p D | Axyz = p D | Axyz = p D | Axyz = p D | Axyz = 316 . If we apply Bayes’s theorem twice more, to take into account Bob’s and Carlos’sopinions given by correlations (3) and (4), using likelihood functions similar tothe one above, we compute the following posterior joint probability distribution, p D | ABCxyz = p D | ABCxyz = p D | ABCxyz = p D | ABCxyz = 0 ,p D | ABCxyz = p D | ABCxyz = 768 , and p D | ABCxyz = p D | ABCxyz = 2768 . Finally, from the joint, we can compute all the moments, including the triplemoment, and obtain E ( XYZ ) = 0 .It is interesting to notice that the triple moment from the posterior is thesame as the one from the prior. This is no coincidence. Because the revisions fromBayes’s theorem only modify the values of the correlations, nothing is changed7 . NEGATIVE PROBABILITIES with respect to the triple moment. In fact, if we compute Deanna’s posteriordistribution for any values of the correlations ǫ A , ǫ B , and ǫ C , we obtain thesame triple moment, as it comes solely from Deanna’s prior distribution. Thus,the Bayesian approach, though providing a proper distribution for the atoms,does not in any way provide further insights on the triple moment. We now want to see how we can use negative probabilities to approach the in-consistencies from Alice, Bob, and Carlos. The first person to seriously considerusing negative probabilities was Dirac in his Bakerian Lectures on the physicalinterpretation of relativistic quantum mechanics [25]. Ever since, many physi-cists, most notably Feynman [26], tried to use them, with limited success, todescribe physical processes (see [27] or [28] and references therein). The mainproblem with negative probabilities is its lack of a clear interpretation, whichlimits its use as a purely computational tool. It is the goal of this section toshow that, at least in the context of a simple example, negative probabilities canprovide useful normative information.Before we discuss the example, let us introduce negative probabilities in amore formal way . Let us propose the following modifications to Kolmogorov’saxioms. Definition 1.
Let Ω be a finite set, F an algebra over Ω , p and p ′ real-valuedfunctions, p : F → R , p ′ : F → R , and M − = P ω i ∈ Ω | p ( { ω i } ) | . Then ( Ω, F , p ) is a negative probability space if and only if:A. ∀ p ′ M − ≤ X ω i ∈ Ω | p ′ ( { ω i } ) | ! B. X ω i ∈ Ω p ( { ω i } ) = 1 C. p ( { ω i , ω j } ) = p ( { ω i } ) + p ( { ω j } ) , i = j. Remark 1.
If it is possible to define a proper joint probability distribution, then M − = 0 , and A-C are equivalent to Kolmogorov’s axioms.Going back to our example, we have the following equations for the atoms. p xyz + p xyz + p xyz + p xyz + p xyz + p xyz + p xyz + p xyz = 1 , (7) p xyz + p xyz + p xyz + p xyz − p xyz − p xyz − p xyz − p xyz = 0 , (8) p xyz + p xyz − p xyz + p xyz − p xyz + p xyz − p xyz − p xyz = 0 , (9) p xyz + p xyz + p xyz − p xyz − p xyz − p xyz + p xyz − p xyz = 0 , (10) p xyz − p xyz − p xyz + p xyz − p xyz − p xyz + p xyz + p xyz = 0 , (11) We limit our discussion to finite spaces. . NEGATIVE PROBABILITIES p xyz − p xyz + p xyz − p xyz − p xyz + p xyz − p xyz + p xyz = − , (12) p xyz + p xyz − p xyz − p xyz + p xyz − p xyz − p xyz + p xyz = − , (13)where (7) comes from the fact that all probabilities must sum to one, (8)-(10)from the zero expectations for X , Y , and Z , and (11)-(13) from the pairwisecorrelations. Of course, this problem is underdetermined, as we have seven equa-tions and eight unknowns (we don’t know the unobserved triple moment). Ageneral solution to (7)-(10) is p xyz = − p xyz = − − δ, (14) p xyz = p xyz = 316 , (15) p xyz = p xyz = 516 , (16) p xyz = − p xyz = − δ, (17)where δ is a real number. From (14)–(17) it follows that, for any δ , someprobabilities are negative. First, we notice that we can use the joint proba-bility distribution to compute the expectation of the triple moment, which is E ( XYZ ) = − − δ. Since − ≤ E ( XYZ ) ≤ − , it follows that − ≤ δ ≤ .Of course, δ is not determined by the lower moments, as we should expect, butaxiom A requires M − to be minimized. So, to minimize M − , we focus only onthe terms that contribute to it: the negative ones. To do so, let us split theproblem into several different sections. Let us start with δ ≥ , which gives M − δ ≥ = − − δ, having a minimum of − when δ = 0 . For − / ≤ δ < , M −− ≤ δ< = δ − + δ = − , which is a constant value. Finally, for δ < − / ,the mass for the negative terms is given by M − δ< − = − δ. Therefore, negativemass is minimized when δ is in the following range − ≤ δ ≤ . Now, going back to the triple correlation, we see that by imposing a minimizationof the negative mass we restrict its values to the following range: − ≤ E ( XYZ ) ≤ . But equations (7)-(13) and the fact that the random variables are ± -valuedallow any correlation between − and , and we see that the minimization ofthe negative mass offers further constraints to a decision maker.Before we proceed, we need to address the meaning of negative probabil-ities, as well as the minimization of M − . We saw from Remark 1 that when M − is zero we obtain a standard probability measure. Thus, the value of M − is a measure of how far p is from a proper joint probability distribution, andminimizing it is equivalent to asking p to be as close as possible to a proper9 . CONCLUSIONS joint, while at the same time keeping the marginals. This point in itself shouldbe sufficient to suggest some normative use to negative probabilities: a negativeprobability (with M − minimized) gives us the most rational bet we can makegiven inconsistent information. But the question remains as to the meaning ofnegative probabilities.To give them meaning, let us redefine the probabilities from p to p ∗ suchthat p ∗ ( { ω i } ) = 0 when p ( { ω i } ) ≤ . It follows from this redefinition that P ω i ∈ Ω p ∗ ( { ω i } ) ≥ . This newly defined probability would not violate Kol-mogorov’s nonnegativity axiom, but instead would violate B above. The p ∗ ’scorresponds to de Finetti’s upper probability measures, and axiom A aboveguarantees that such upper is as close to a proper distribution as possible. Thus,according to a subjective interpretation, the negative probability atoms corre-spond to impossible events, and the positive ones to an upper probability mea-sure consistent with the marginals. Once again, the triple moment correspondsto our best bet. The quantum mechanical formalism has been successful in the social sciences.However, one of the questions we raised elsewhere was whether some minimal-ist versions of the quantum formalism which do not include a full version ofHilbert spaces and observables could be relevant [19]. In this paper we adaptedthe example modeled with neural oscillators in [19] to a different case where eachrandom variable could be interpreted as outcomes of a market, and where theinconsistencies between the correlations could be interpreted as inconsistenciesbetween experts’ beliefs. Such inconsistencies result in the impossibility to definea standard probability measure that allows a decision-maker to select an expec-tation for the triple moment. The computation of the triple moment from theinconsistent information was done in this paper using three different approaches:Bayesian, quantum-like, and negative probabilities.With the Bayesian approach, we showed that not only does it rely on amodel of the experts (the likelihood function), but also that no new informationis gained from it, as the triple moment from the prior is not changed by theapplication of Bayes’s rules. Therefore, the Bayesian approach had nothing tosay about the triple moment.Similar to the Bayesian, the quantum approach also had nothing to say aboutthe triple moment, as the arbitrariness of choices for quantum superpositions(without any additional constraints) results in all values of triple moments beingpossible. In fact, the quantum approach above could be similarly implementedusing a contextual theory. For instance, Dzhafarov [29] proposes the use of anextended probability space where different random variables (say, X z and X y )are used, and where we then ask how similar they are to each other (for instance,what is the value of P ( X z = X y ) ). However, as with the quantum case, themeaning given to P ( X = 1) in our example does not fit with this model, as itcorresponds to the expectation of an increase in the stock value of company X . CONCLUSIONS in the future, and the X that Alice is talking about is exactly the same onefor Bob and Carlos, as it corresponds to the increase in the objective value (inthe future) of a stock in the same company. Furthermore, as expected due toits similar features, this approach has the same problem as the quantum one interms of dealing with the triple moment, but it has the advantage of making itclearer what the problem is: the triple moment does not exist because we havenine random variables instead of three, as we have three different contexts.The negative probability approach, on the other hand, led to a nontrivialconstraint to the possible values of the triple moment. When used as a computa-tional tool, a joint probability distribution, and with it the triple moment, couldbe obtained. Together with the minimization of the negative mass M − , this jointleads to a nontrivial range of possible values for the triple moment. Given theinterpretation of negative probabilities with respect to uppers, it follows thatthis range is our best guess as to where the values of the triple moment shouldlie, given our inconsistent information. Thus, negative probabilities provide thedecision maker with some normative information that is unavailable in eitherthe Bayesian or the quantum-like approaches. Acknowledgments.
Many of the details about negative probabilities were devel-oped in collaboration with Patrick Suppes, Gary Oas, and Claudio Carvalhaeson the context of a seminar held at Stanford University in Spring 2011. I amindebted to them as well as the seminar participants for fruitful discussions. Ialso like to thank Tania Magdinier, Niklas Damiris, Newton da Costa, and theanonymous referees for comments and suggestions.
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