On the calculus of subjective probability in behavioral economics

49Asian Journal of Economics and Banking (2020), 4(1), 49–60 Asian Journal of Economics and Banking ISSN 2615-9821 On the Calculus of Subjective Probability in Behavioral Economics Hung T. Nguyen„ New Mexico State University (USA) & Chiang Mai University (Thailand) Article Info Received: 06/02/2020 Accepted: 16/3/2020 Available online: In Press Keywords Bayesian Probability, Belief Functions, Choquet Ca- pacity, Coarse Data, Frequen- tist Probability, Fuzzy Sets, Granularity, Idempotent Analy- sis, Linguistic Variables, Possi- bility Distributions, Possibility Measures, Random Sets, Sub- jective Probability. JEL classification C10, C11 MSC2020 classification 60A05, 62C10 Abstract In elaborationg upon the recent thought-provoking paper“Subjective probability in behavioral economics and finance: A radical reformulation” by H. Joel Jef- frey and Anthony O. Putman [5], we proceed to spec- ify the calculus of their “probability (uncertainty) ap- praisals” as possiblity measures, i.e., the “radical” re- formulation of the usual calculus of subjective proba- bilities is that of idempotent uncertainty. With pos- sibility measures as quantitative uncertainty for sub- jective probabilities, we discuss the necessary mariage of possibility measures and Kolmogorov probability measures in a new Bayesian analysis for economic ap- plications. „Corresponding author: hunguyen@nmsu.edu 50 Asian Journal of Economics and Banking (2020), 4(1), 49–60 1 INTRODUCTION As stated in the abstract, this paper is about the recent thought-provoking paper [5]. It is thought-provoking be- cause it proposed a “radically new” way to really understand the old notion of subjective probability, from which ap- plications, e.g., in decision-making in social sciences, will be “radically differ- ent”, say, more “compatible” with what psychological experiments revealed (in behavioral econometrics and finance). As such, it is important to expose the essentials of [5] to a large audi- ence of econometricians. In fact, by do- ing so, we accomplish several important tasks, namely specifying the calculus of subjective probabilities as (idempotent) possibility as previously suggested [2], and discussing the necessary mariage of possibility calculus with standard Kol- mogorov probability calculus, which is needed in a “Bayesian” framework of statistical inference. Since the “beginning”, we are taught that, in constrast to frequentist prob- ability (Kolmogorov), subjective (or Bayesian) probabilities are in our mind, which are used as a mode of judge- ment. When using subjective proba- bilities in applications, such as in auc- tions (as Bayesian games) or Bayesian statistics, not only we need to manip- ulate quantitative subjective probabil- ity, but also combine subjective prob- ability with frequentist probability (to form Bayesian statistics). While, the mathematical (language) foundation for quantitative (frequen- tist) probability is Kolmogorov’s axioms (by analogy with measure theory), in- cluding additivity, what are the axioms, i.e., the calculus of subjective probabil- ity, that the Bayesians use to establish their Bayesian statistical theory? Can you guess? “Imagine” if the cal- culus of subjective probabilities is dif- ferent than Kolmogorov probability cal- culus! Then how can Bayesian anal- ysis be carried out? Specifically, how to incorporate prior information into a frequentist framework? Well, we know that, as stated again in [5], the stan- dard “view” is that with respect to cal- culus (of uncertainties), frequentist and subjective probabilities are two sides of the same coin, i.e., while their meanings are different, their calculi are the same. This allows Bayesian statistics to exist to “beat” frequentist statistics in several fronts, e.g., in hypothesis testing. I can’t resist to point out an op- posite situation: while the meaning of the (intrinsic) uncertainty in quan- tum mechanics is the same as that of ordinary randomness, their calculi are different, namely, quantum probabil- ity is non-commutative whereas Kol- mogorov probability is not (but, in fact, quantum probability is simply a non-commutative generalization of Kol- mogorov probability). Should we ask “why it is so?”. More specifically, “Why subjective probabil- ities are additive, or even σ−additive, just like Kolmogorov probabilities?”. Well, texts on subjective probability ex- plain that, by using betting schemes, they are so by existence of Dutch book arguments for these properties. Note that, although Bertrand Rus- sel once wrote in 1929 “Probability is the most important concept in mod- Hung T. Nguyen/On the Calculus of Subjective Probability in Behavioral Economics 51 ern science, especially as nobody has the slighest notion what it means”, peo- ple equate uncertainty with probabil- ity whose calculus is Kolmogorov. Thus, “traditionally”, while subjective and ob- jective probabilities are different man- ifestations of uncertainty, people ma- nipulate them according to the same calculus. After Dennis Lindley attended a seminar at UC Berkeley in 1981 in which both Lotfi Zadeh and Glenn Shaffer talked about their non-additive uncertainty measures (possibility mea- sures, and belief functions, respec- tively), he wrote [7] claiming that all non-additive uncertainty measures are inadmissible, i.e., the message is “we cannot avoid probability” (where, of course, again, by“probability”, we mean additive set functions). It turns out that Lindley’s message is not exactly what he claimed! What he did show is that “Us- ing scoring rule approach, an admissible uncertainty measure must be a function of a probability measure”. But then, for example, Shaffer’s belief functions which are non-additive, are functions of probability measures, and hence admis- sible in Lindley’s sense! A complete re- sponse to Lindley’s paper was [4]. One more thing about the coexis- tence of objective uncertainty (say, in von Neumann-Morgenstern utility the- ory) and subjective uncertainty (say, in Savage’s subjective/qualitative prob- ability theory): this is possible in appli- cations since these two different types of uncertainty are forced to obey the same calculus. For a rigorous treatment, see [6], where it reminded the reader that “One warning: when mathematicians use the term probability , they almost always mean a σ−additive probability measure defined on a σ−algebra”! It is clear in [6] that uncertainty theo- ries are proposed to be used in mod- eling behavior of individuals in their decision-making, and as such, e.g., in physics, models of choice must be con- firmed by experimental evidence: they were not (See the last Chapter of [6] on “The Experimental Evidence”). Thus, the door was open ever since for non-additive uncertainty measures. It should be emphasized for statisticians that Kolmogorov probability is just one quantitative modeling of one type of un- certainty. There are other types of un- certainty whose modelings might not be “probability”, i.e., not additive. Do not equate uncertainty with probabil- ity. How to find out reasonable quan- titative theories of uncertainty? Well, just follow physics (I mean quantum mechanics)! Quantitative modeling of a type of uncertainty is used to model the behavior of something, and as such, a proposed model must be tested by ex- periments. The non-commutativity and non-additivity of probability in quan- tum mechanics, as observed in experi- ments, led to the establishment of a firm theory of quantum probability. In so- cial sciences, including economics and fi- nance, people make decisions under un- certainty. Thus, it is so clear that any quantitative theory of uncertainty must be validated by experimental evidence. The thought-provoking paper under re- view is precisely in this scientific spirit. Somewhat clearly (!) that statisti- cians and econometricians are still not aware of non-additive set functions which were proposed to model various 52 Asian Journal of Economics and Banking (2020), 4(1), 49–60 different types of uncertainty we face in everyday decision-making, say in Arti- ficial Intelligence, (not just one type of uncertainty, traditionally attributed to randomness or epistemic uncertainty, as two sides of the same coin, i.e., obey- ing the calculus of Kolmogorov proba- bility theory), let alone fuzzy set the- ory of Zadeh (1965, see [10] for a com- plete update of the theory) from which are founded concepts such as approxi- mate reasoning, soft computing, gran- ular computing, possibility theory (see e.g., [1,3,13–16]). As we will see shortly, the“radical reformulation”of the notion of subjective probability (uncertainty) in [5] is “probability appraisals” which are nothing else than Zadeh’s “linguis- tic variables” [17] in the context of fuzzy set theory. In this context, it looks like we are heading back into the territory of fuzzy set theory which, since 1965, only attracted computing and engineer- ing fileds. Perhaps it will be so since, af- ter all, there are “real” contributions of fuzzy theory to social sciences, includ- ing economics and finance. The purpose of this present paper is multifold: First, we elaborate of the “radical reformulation” of [5] which we believe that it was in a right direction for improving methods in behavioral economics and finance. Next, we place this reformulation completely in the set- ting of linguistic variables and granu- lar information. Then, we complete [5] by specifying a reasonable calculus of subject probabilities, namely possibil- ity measures (as opposed to probabil- ity measures). Finally, we discuss a “Bayesian” analysis in which possibil- ity uncertainty coexists with probability uncertainty. 2 UNCERTAINTY APPRAISALS In a sense, as opposed to objective uncertainty (quantitatively modeled as frequentist probabilities), by subjective uncertainty we mean the type of un- certainties (e.g., epistemic uncertainty) which cannot be modeled quantitatively by a frequentist approach (e.g., for events which cannot be repeated), it is said that, for such uncertainty, we judge it by using our mind. But how exactly our mind perceives it, let alone manipu- lates it (i.e., what is the calculus of sub- jective probabilities?). Well, you might say : it’s an old story and it has been resolved long time ago! Note that, if we adapt the current Bayesian calcu- lus of subjective probabilities, then fre- quentist and Bayesian statistics are just two sides of the same coin (in the sense that they use the same calculus of prob- abilities). While it has been known, also for a long time, from experimental evi- dence, that humans do not necessarily manipulate their subjective “probabili- ties” (uncertainties) according to Kol- mogorov calculus, the paper [5] seems to spell it out specifically as “uncer- tainty appraisals” which could lead to a “radical” calculus of subjective uncer- tainty. We elaborate next the main mes- sage in [5], namely“Subjective probabil- ities are uncertainty appraisals” in so- cial decision-making context, with em- phasis on Bayesian statistics framework in econometrics. When facing, say, an epistemic un- certainty (e.g., on an unknown parame- Hung T. Nguyen/On the Calculus of Subjective Probability in Behavioral Economics 53 ter of a population), we take a “closer look” at it, then use any information we have about it to “appraise” it, i.e., how to describe the uncertainty a lit- tle more precise for, say, actions in a decision-making. Of course, as in gen- eral thinking processes, humans tend to use natural languages before numerical languages. As such, an appraisal of an uncertain situation should be a linguis- tic variable in Zadeh’s sense [17], i.e., a variable whose possible values are words in a natural language. If the variable X is an appraisal of uncertainties, then its possible set of values could be “likely, very likely, unlikely, ....” which can be modeled as fuzzy subsets of the unit in- terval [0, 1], i.e., fuzzy probabilities. In this quantitative modeling process, the subjective aspect of the appraisals is reflected in the shapes of membership functions of fuzzy probabilities. When trying to model, say, the epis- temic uncertainty concerning an un- known parameter θ of a population (as in standard practice of Bayesian statis- tics), we view θ as a linguistic vari- able instead. The parameter space Θ (containing the true parameter) can be coarsen into a fuzzy partition to provide granular information about θ, i.e., a set of possible linguistic values for θ, from which we could consider granular infor- mation of the form “θ is A is λ”, where A is a fuzzy subset of Θ, and λ is a fuzzy probability (e.g., “θ is “small” is likely”). In order to “figure out” how to ma- nipulate (in other words, how to derive a calculus of) uncertainty appraisals, it is necessary to dig into the model- ing of membership functions of fuzzy “probabilities”. Roughly speaking, say- ing that something is, e.g., likely, is say- ing that it is “possible” it is so. We used to hear statements such as “some- thing is improbable but possible”, re- vealing that possibility is a weaker no- tion than probability. Moreover, it seems that a quantitative theory of pos- sibility was first systematically formu- lated by Zadeh [14]. See an update in [1]. In the next section, we will pro- ceed to advocate that the qualitative approach to uncertainty appraisal in [5] could be quantitatively formulated by Zadeh’s possibility theory. 3 POSSIBILITY MEASURES Probability theory (Kolmogorov) is a quantitative theory of objective uncer- tainty. The meaning of this uncertainty is based on the notion of frequency, and its calculus (i.e., the way probabilities are combined) is based upon an anal- ogy with measure theory in mathemat- ics (with its main axiom of additiv- ity). There are various different types of uncertainty, e.g., subjective probabil- ity, fuzziness. When trying to establish- ing a quantitative theory for a kind of uncertainty, we first examine carefully the meaning of the uncertainty under consideration, then from which, proceed to postulate its calculus (i.e., axioms). This is exemplified by Kolmogorov’s for- mulation of (objective) probability the- ory. Now, the very reason that we are concerned with uncertainty is decision- making: how we make decisions in the face of a type of uncertainty? The an- swer to this question dictates an appro- priate calculus for that type of uncer- 54 Asian Journal of Economics and Banking (2020), 4(1), 49–60 tainty. It is “interesting” to ask whether we should have an axiomatic theory of uncertainty in advance or derive it from experimental evidence? Suppose in natural sciences, uncer- tainty is objective so that we could use the frequentist viewpoint to model uncertainty as (Kolmogorov) probabil- ity. Does that mean that its calculus must always obey Kolomogorov’s calcu- lus? Well, as we all know, one century ago, that was not true in quantum me- chanics: while the meaning of probabil- ity is the same, the calculus of quan- tum probability is different than that of Kolmogorov (e.g., non additive and non commutative). Note however that quantum probability calculus is a gener- alization of Kolmogorov probability cal- culus. Perhaps following physics, social sci- entists have looked at various types of uncertainties encountered in social problems since quite some time, reveal- ing various different calculi of uncertain- ties (See [6] for a survey). Now, it is well known that there is a type of uncertainty, called epistemic uncertainty in Bayesian statistics. The quantitative modeling of this uncer- tainty is called “subjective probability” because of the following reasons. First of all, it is subjective! The “probability” of an uncertain phenomenon is varies from person to person, and is used to quantify the uncertainty to make deci- sions. Secondly, it is only in your mind. And, finally, it is called “probability” since its calculus (axioms) is taken to be exactly Kolmogorov’s calculus, despite their difference in semantics (subjective probabilities are not frequentist-based). We have pointed out earlier why we are led to a situation like this! Is it the time to take a “closer look” at the calculus of subjective probabil- ities? say, in the “spirit” of quantum physics! Here, we examine [5] and try to figure out, in view of their “radical re- formulation”of subjective probability in behavioral economics and finance, what should be a new and reasonable calculus for subjective probabilities? Since possibility measures, as rea- sonable quantitative candidates for “subjective probabilities” in social sci- ences, seem new to social scientists, we devote this entire section to a tutorial on how the intuitive (and well-known) notion of possibility arises in scientific studies. Upfront, as stated earlier, we are going to justify Zadeh’s possibility theory. Possibility theory is a mathe- matical theory of a weaker form of un- certainty (possibility). While the in- tuitive concept of possibility is famil- iar in everyday language, it has not been systematically used as a“scientific” tool (like probability) in, say, decision- making. Now, as mentioned above, the “acceptable” notion of “subjective prob- ability” (as Bayesians call it) seems to resemble to possibility rather than prob- ability (in [5], subjective probability is not probability!), it is time to make it clear: What is the calculus of subjec- tive probability? i.e., how possibility measures are axiomatized? (just like objective probability measures are ax- iomatized by Kolmogorov, from which, their calculus follows). In the language of Bayesian statistics, we are basically talking about the “prior information” of the true but unknown “parameter” θo in Hung T. Nguyen/On the Calculus of Subjective Probability in Behavioral Economics 55 the parameter space Φ. We are going to call it a “possibility distribution” of θo, i.e., a function pi(.) : Θ→ [0, 1] with supθ∈Θ pi(θ) = 1, where pi(θ) represents the possibility (a value in [0, 1]) that θ is the true parameter. From pi(.), the notion of possibility that the true pa- rameter is in a subset A ⊆ Θ is “postu- lated” to be pi(A) = supθ∈A pi(θ). This is Zadeh’s theory of possibility measures [16]. We are going to justify the Zadeh’s calculus of possibilities. Just like “sub- jective probabilities”, possibilities are somewhat subjective and are used to specify“appraisals”for decision-making. They can come from your mind (in fact, based on your available information). Now, in everyday language, probability is often used to strengthen possibility, so that there is some “intuitive” link be- tween these two notions of uncertainty. While, once the meaning of possibilities is spelled out, their calculus could be de- rived from the so-called “experimental evidence”, we offer here a justification based on the intuitive link of probabil- ity and possibility. Probability and possibility are some- what related? We often say “that some phenomenon is improbable but possible”, meaning that possibility is a weaker degree of belief. Typically, such statement appears in situations where we face coarse data (imprecise or low quality data). Let (Ω,A, P ) be a prob- ability space. Let Θ be a set. Sup- pose when we perform an experiment, we observe only the outcome in a sub- set of Θ (rather than a precise value in it), i.e., our random experiment is de- scribed by a random set S(.) : Ω→ 2Θ, so that our random variable of inter- est X(.) : Ω → Θ is an a.s. selec- tor of S. An event A ⊆ Θ is realized (occured) if X(ω) ∈ A, but if we only know that X(ω) ∈ S(ω), what can we say? Well, if S(ω) ⊆ A, then A occurs, but if S(ω) ∩ A 6= ∅, then it is possi- ble that A occurs. Thus, we could take pi(A) = P (ω : S(ω) ∩ A 6= ∅) to quan- tify the possibility that the event A oc- curs, i.e., the possibility measure of the event A. Since X is an almost sure se- lector of S, we have {ω : X(ω) ∈ A} ⊆ {ω : S(ω) ∩ A 6= ∅}, and hence P (ω : X(ω) ∈ A) ≤ P (ω : S(ω) ∩ A 6= ∅), i.e., “probability is smaller than possi- bility”, as expected. Let A = {θ} be a singleton set, then the possibility of θ is pi(θ) = P (ω : S(ω) 3 θ), which is the coverage function of the random set S. Thus, formally, a possibility dis- tribution is the coverage function of a random set. The question now is this. What is the relation between pi(θ) and pi(A)? Of course, you are thinking about a possi- ble analogy with probability theory in which pi(θ) plays the role a a probability denstity function, whereas pi(A) plays the role of a probability measure. Now, let pi(.) : Θ → [0, 1] such that supθ∈Θ pi(θ) = 1. Then there is a ran- dom set S(.) on Θ such that P (ω : S(ω) ∩ A 6= ∅) = supθ∈A pi(θ), for each A ⊆ Θ. In other words, there is a canonical random set S whose coverage function is precisely the possibility dis- tribution function pi(.), and the possi- bility measure of any event A ⊆ Θ is given as supθ∈A pi(θ). Such a result jus- tifies Zadeh’s postulates pi(.) : Θ ∈ [0, 1] with supθ∈Θ pi(θ) = 1, and pi(A) = 56 Asian Journal of Economics and Banking (2020), 4(1), 49–60 supθ∈A pi(θ), noting that pi(∅) = 0. Here is the proof. Let α(.) : (Ω,A, P ) → [0, 1], uniformly dis- tributed. Consider the random set S(.) : (Ω,A, P ) → 2Θ which is the random- ized level set of the function pi(.) : Θ→ [0, 1], i.e., S(ω) = {θ ∈ Θ : pi(θ) ≥ α(ω)}, Then, for A ⊆ Θ, we have pi(A) = P (ω : S(ω) ∩ A 6= ∅) = P{ω : α(ω) ≤ sup θ∈A pi(θ)} = sup θ∈A pi(θ), Remark. From pi(A) = supθ∈A pi(θ), we see that, in particular, the possibility measure pi(.) on 2Θ, is maxitive (rather than additive), i.e., for any A,B, we have pi(A∪B) = max{pi(A), pi(B)}.It is interesting to note that maxitive opera- tor can arise from additive operator in some limiting process. Let Pn, n ≥ 1 be a sequence of probability measures on a measurable space (Ω,A), and εn > 0, with εn → 0 as n→∞. Then it is pos- sible that the sequence of submeasures P εnn , n ≥ 1 converges to a (σ−) maxitive set-function. Here is an example. Let (Ω,A, P ) be a probability space, and f(.) : Ω→ R+, f ∈ L∞(Ω,A, P ). Since P is a finite measure, it follows that f ∈ Lp(Ω,A, P ), for all p > 0. Con- sider Pn(A) =
A |f(ω)|ndP
Ω |f(ω)|ndP , then, as n→∞, [Pn(A)] 1 n = ||f1A||n ||f ||n → ||f1A||∞ ||f ||∞ , where the set function pi(A) = ||f1A||∞ ||f ||∞ is (σ−) maxitive. Of course, if we“consider”a function pi(.) : Θ → [0, 1] with supθ∈Θ pi(θ) = 1, and call it our possibility distribution (just like “probability density”, where “possibility” reflects our “subjective” as- signments/ appraisals), then our possi- bility measure of A ⊆ Θ is taken to be pi(A) = supθ∈A pi(θ), although we do not need to relate it to the notion of probability. In other words, uncer- tainties might not need to be a function of probability. For example, a function pi(.) as above plays the role of a mem- bership function of a fuzzy subset of Θ, generalizing ordinary subsets of Θ (see [15] for more discussions on how to ob- tain possibility distributions in applica- tions). What we have done above is to give a meaning to the concept of possi- bility distribution (from which possibil- ity measures are built), namely“a possi- bility distribution is the coverage func- tion of a random set”. If you refer to the concept of confidence intervals in statis- tics, then you realize the flavor of “sub- jective” assessments. It is interesting to note that, for- mally, the concept of possibility dis- tribution is somewhat “hidden” within probability theory! In fact, the notion of possibility distribution“appears”also in probability theory as limits of probabil- ities. When X is an absolutely continu- ous random variable, e.g., taking values, say, in R = (−∞,∞), with distribution function F (x) = P{ω : X(ω) ≤ x}, then its density function value Hung T. Nguyen/On the Calculus of Subjective Probability in Behavioral Economics 57 f(x) = lim h→0 F (x+ h)− F (x− h) 2h = lim h→0 P{ω : x− h ≤ X(ω) ≤ x+ h}) 2h can be interpreted as a possibility value (and not a probability value). For ex- ample, in the context of Bayesian statis- tics, when the parameter space is Θ = [0, 1], the uniform prior density f(θ) = 1, for any θ ∈ [0, 1] is a possibility dis- tribution, resulting in supθ∈A f(θ) = 1 for any A ⊆ [0, 1]. And this can be applied to unbounded parameter spaces like R = (−∞,∞) with a possibility distribution f(θ) = 1, for any θ ∈ R, without “calling” it an improper prior probability density function ! To be complete, we mention here that possibility measures can be also in- terpreted as limits of probability mea- sures in the large deviation convergence sense (See [8], [11]). Recall that the purpose of the study of large deviations in probability theory is to find the asymptotic rate of conver- gence of sequences of probability mea- sures of rare events. For example, in the simplest setting, while the law of large numbers asserts that the sequence of sample means (of i.i.d. Xj, j ≥ 1, with EX = µ, V ar(X) = σ2 < ∞) Xn con- verges surely to µ, it is also of interest to find the rate at which P (|Xn − µ| > a) (probability of large deviations from the mean) goes to zero, as n→∞, for some a. Now, for Zn = √ n(Xn − µ)/σ, we have, by the Central Limit Theorem, P (Zn > x) ≈  ∞ x 1√ 2pi e− t2 2 dt By finding the asymptotics of this integral for x → ∞, we arrive at the well-known Cramer result of the form : The sequence of sample means satisfies the large deviation principle in the sense that, for ε > 0, we have P (|Xn − µ| > a) ≤ e−nh(a)+ε for n sufficiently large, where the ex- ponential rate of convergence being e−nh(a). The general setting of large devi- ation principle is this. Let Θ be a complete, separable metric space, and B its borel σ−field. A sequence of probability measures Pn on (Θ,B) is said to obey the large deviation prin- ciple (LDP) if there exists a lower semi- continuous function I(.) : Θ → [0,∞] (the rate function) such that : (i) for each closed set F of Θ, lim sup n→∞ [ 1 n logPn(F )] ≤ − inf θ∈F I(θ) (ii) for each open set G of Θ, lim sup n→∞ [ 1 n logPn(G)] ≥ − inf θ∈G I(θ) If we let ϕ(A) = supθ∈A e −I(θ), then the LDP is formulated as lim sup n→∞ [Pn(F )] 1 n ≤ ϕ(F ) and lim sup n→∞ [Pn(G)] 1 n ≤ ϕ(G) 58 Asian Journal of Economics and Banking (2020), 4(1), 49–60 If we refer the above to the Port- mamteau theorem of weak convergence of probability measures, we can view the set function ϕ(.) as a limit of Pn in the sense of LDP. Remark. Since the possibility dis- tribution ϕ(θ) = e−I(θ) is upper semi- continuous (with values in [0, 1]), the associated possibility measure ϕ(.) char- acterizes the probability distribution of a random closed sets on Θ (See [9] for details). In summary, a possibility distribu- tion (for an appraisal) is a function pi(.) on an arbitrary set Θ, taking val- ues in the unit interval [0, 1], such that supθ∈Θ pi(θ) = 1. Its associated possi- bility measure, denoted also as pi(.), is a map from the power set 2Θ of Θ to [0, 1], defined by pi(A) = supθ∈A pi(θ), for A ⊆ Θ. Equivalently, a possibility measure is a map pi(.) : 2Θ → [0, 1] such that pi(∅) = 0, pi(Θ) = 1, and pi(∪i∈IAi) = supi∈I pi(Ai), for any index set I. Remark. For a general theory of idempotent probability, see [11]. 4 BAYESIAN ANALYSIS WITH POSSIBILITY MEASURES First of all, in the context of Bayesian statistics, the additional in- formation required is a prior proba- bility measure (or often a probabil- ity density function) on the parame- ter space, reflecting the subjective in- formation about the possible location of the true (but unknown) parameter of the population. Such an informa- tion is quite compatible with a possibil- ity distribution where the value of the possibility distribution of an element in the parameter space acts as the degree of membership of that element in the statistician’s perception. Secondly, while the literature did contain research works on the possi- bility to consider “Bayesian statistics” where both prior and objective proba- bility measures are replaced by possi- bility measures, it seems that the more practical situation where the prior sub- jective (epistemic) uncertainty is pos- sibilistic, but the objective uncertainty (in the statistical models) remains Kol- mogorov is not completely worked out. This is the situation where two different types of uncertainty calculi need to be combined! The paper [5] is, in a sense, a con- firmation of [2], from an original idea in [16]. And now, with all necessary justifications spelled ont in the previous section, it is about time to see how “tra- ditional”Bayesian statistics is affected?! This will happen clearly when we simply replace the calculus of subjective prob- abilities by the calculus of possibilities in the formulation of the “Bayes the- orem”, combining objective probability with subjective probability. A situation closely related to what we wish to discuss here is robust Bayesian statistics, as exemplified by [12]. In a sense, robust Bayesian in- ference refers to the situation where we consider a set of possible priors (say, as prior probability measures on the pa- rameter space) rather just one given one. Specifically, let P be a set of prob- ability measures on (Θ,B(Θ)). With- out knowing a specific prior in P , we must work with the lower and upper en- Hung T. Nguyen/On the Calculus of Subjective Probability in Behavioral Economics 59 velops of P , i.e., the non-additive set- functions L(.) = infP∈P P (.), U(.) = supP∈P P (.), respectively. However, for each P ∈ P , we can use Bayes theorem to get the corresponding posterior prob- ability measure. Bounds on these pos- terior probability measures can be ob- tained from L(.) and U(.). Specifically, as in [12], these bounds are expressed as (Choquet) integrals of monotone in- creasing set functions (called Choquet capacities) L(.) and U(.), as a general- ization of ordinary integral calculus. Now, in one hand, possibility mea- sures are Choquet capacities, and on the other hand, if we view a possibil- ity measure pi(.) as the envelop of a set of prior probability measures P , i.e., say, P = {P : P (.) ≤ pi(.)}, then we can proceed as in [12], noting that a possibility measure pi(.) is , in par- ticular, maxitive, i.e., for A,B in 2Θ, pi(A∪B) = max{pi(A), pi(B)}. But, any such set-function is “alternating of infi- nite order” (and hence, in particular, 2- alternating), i.e., for any A1, A2, ..., An , we have pi(∩ni=1Ai) ≤ ∑ ∅ 6=I⊆{1,2,...,n} (−1)|I|+1pi(∪i∈IAi) where |I| denotes the cardinality of the set I. For a proof, see [9]. In general, the problem seems open. References [1] Dubois, D., & Prade, H. (2011). Possibility theory and its applications: Where do we stand? Mathware and soft computing, 18(1), 18–31. [2] Dubois, D. (2006). Possibility theory and statistical reasoning. Computational Statistics and Data Analysis, 51, 47-69. DOI:10.1016/j.csda.2006.04.015. [3] Goodman, I. R., & Nguyen, H. T. (1985). Uncertainty Models for Knowledge- Based Systems, North-Holland [4] Goodman, I. R. , Nguyen, H. T., & Rogers, G. S. (1991). On the scoring approach to admissibility of uncertainty measures in expert systems, J. Math. Anal. and Appl., 159, 550–594. [5] Jeffrey, H. J., & Putman, A. O. (2015). Subjective probability on behavioral economics and finance: A radical reformulation, J. Behavioral Finance, 16, 231–249. [6] Kreps, D. V. (1988). Notes on The Theory of Choice. Westview Press. [7] Lindley, D. V. (1982). Scoring rules and the inevitability of probability. Intern. Statist. Review, 50(1), 1–11. [8] Nguyen, H. T., & Bouchon-Meunier, B. (2003). Random sets and large devi- ations principle as a foundation for possibility measures. Soft Computing, 8, 61–70. 60 Asian Journal of Economics and Banking (2020), 4(1), 49–60 [9] Nguyen, H. T. (2006). An Introduction to Random Sets. Chapman and Hall / CRC Press. [10] Nguyen, H. T., Walker, L. C., & Walker, E. A. (2019). A First Course in Fuzzy Logic (Fourth Edition). Chapman and Hall / CRC Press. [11] Puhalskii, A. (2001) Large Deviations and Idempotent Probability. Chapman and Hall / CRC Press. [12] Wasserman, L. A., & Kadane, J. B. (1990). Bayes’ theorem for Choquet ca- pacities. Ann. Statist., 18(3), 1328–1339. [13] Yao, J. T., Vasilakos, A. V., & Pedrycz, W. (2013). Granular computing: Perspectives and challenges. IEEE Trans Cybern., 34(6), 1977–1989. [14] Yao, M. X. (2019). Granularity measures and complexity of partition-based granular structures. Knowledge-Based Systems, 163, 885–897. [15] Zadeh, L. A. (1996). Fuzzy sets and information granularity. In G.J. Klir and B. Yuan (Eds), Fuzzy Sets, Fuzzy Logic and Fuzzy Systems (Selected papers of Lotfi A. Zadeh) (433-448). World Scientific. [16] Zadeh, L. A. (1978). Fuzzy sets as a basis for a theory of possibility. J. Fuzzy Sets and Systems, 1, 3–28. [17] Zadeh, L.A. (1975). The concept of a linguistic variable and its application to approximate reasoning. I (Information Sci., 8, 199-249), II (Infomation Sci. 8, 301-357), III (Information Sci. 9, 43-80)