Abstract
The subjectivist approach permits to apply probability calculus to the greatest possible number of feelings of uncertainty, when one takes the view that it can be defined only for a specific individual and not for an event as in the objectivist one. It uses the notion of coherence in individual behavior reconciled with the notion of utility of winning. Different sets of axioms were proposed during the twentieth century for this approach the main one being given by de Finetti. Such an approach using the notion of exchangeability permits a clear answer to the problem of statistical inference. Different examples of application are then given in jurisprudence and in educational science. Finally we present some criticisms of this approach, which is too closely tied to individual psychology.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Omnia, quæ sub sole sunt vel fiunt, præterita, præsentia, præsentia sive futura, in se & objectivè summam semper certitudinem habent.
- 2.
Certitudo rerum, spectatata in ordine at nos, non omnium eadem is, sed multipliciter variat secundùm magis & minus. …Cætera omnia imperfectiorem ejus mensuram in mentibus nostris obtinent, majorem minoremve, prout plures vel pauciores sunt probabilitates, quæ suadent rem aliquam esse, fore aut fuisse.
- 3.
Probabilitas enim is gradus certitudinis, &ab hac differt ut pars à toto.
- 4.
Ita cùm qæritur in abstracto, quantò sit probabilius, juvenem vigenti annorum senem sexagenario fore superstitem, quàm verò hunc illi, præter discrimen ætatis & annorum nihil is, quod considerare possis; sed ubi specialiter sermo is de individuis Petri juvenis & Pauli senis, attendere insuper opportet ad specialem eorum complexionem & studium, quo uterque valetudinem suam curat; nam si Petrus sit valetudinarius, if infectibus indulgeat, if intepemperanter vivat, fieri potest, ut Paulus, etsi ætate provectior, optima tamen ratione longioris spem vitæ concipere valeat.
- 5.
si modo haberi possunt.
- 6.
si ex. gr. facto olim experimento in tercentis hominibus ejusdem, cujus nunc Titius is, ætatis & complexionis, observaveris ducentos eorum ante exactum decennium mortem oppetiisse, reliquos ultravitam protraxisse, satis tu colligere poteris, duplo plures casus esse, quibus & Titio intra decennium proximum naturae debitutm solvendum sit, quàm quibus terminium hunc transgredi possit.
- 7.
This designation allows the principle to be contrasted with Leibniz’s principle of sufficient reason, which posits that for each fact there exists a sufficient reason to explain why it occurs and not another. Keynes, who was dissatisfied with the term, renamed it the indifference principle.
- 8.
omnes casus æquaè possibiles esse, seu pari facilitate evenire posse;
- 9.
pono in urna quadem te inscio reconditos esse ter thousand calculos albos & bis thousand nigros, teque eorum nyumerum experimentis exploraturum educere calculum unum post alternum (reponendo tamen singulis vicibus illum quem eduxisti, priusquam sequentem eligas, ne numerus calculorum in urna minuatur) & observare, quoties albus & quoties ater exeat.
- 10.
binis limitibus conclusam, sed qui tam arcti constitui possunt, quam quis voluerit.
- 11.
This paradox owes its name to the fact that Nicolas Bernoulli’s cousin Daniel Bernoulli published a paper on the problem in the Commentaires de l’Académie des Sciences de Saint-Pétersbourg in 1738.
- 12.
valor non is aestimandus ex pretio rei, sed ex emolumento, quod unusquisque inde capessit. Pretium ex re ipsa aestimatur omnibusque idem is, emolumentum ex conditione personae.
- 13.
Cum emolumenta singula expectata multiplicantur per numerum casuum, quibus obtinetur aggregatumque productorum dividitur per numerum omnium casuum, obtinebitur emolumentum medium, and lucrum huic emolumento respondens aequivalebit sorti quaesitae.
- 14.
Bayes, in fact, seeks the more complex probability that the sought-for probability lies in an interval [b, f]. He thus obtains an integral relative to \( {\widehat{p}}_{n}\), between b and f, of the equation below divided by 2.
- 15.
A number of authors (Pearson 1920; Fisher 1956; Hacking 1965) have not properly understood this hypothesis advanced by Bayes; they believe that it consists of Laplace’s principle of insufficient reason, which we shall examine later. Laplace’s principle states that, when no information exists, it is the unknown probability that we must regard as uniformly distributed. In this case, we would also be unaware of any monotonic function of the unknown probability, which would yield different results (see Stigler (1986) for a fuller discussion of this hypothesis).
- 16.
Interestingly, Laplace does not seem to have been aware of Bayes’s work at that date, for the introduction to his paper (written by Condorcet) does not mention Bayes. By contrast, 4 years later (1778), Laplace’s introduction by Condorcet quotes Bayes and Price, who published his results in the Philosophical Transactions.
- 17.
This hypothesis is therefore different from Bayes’s hypothesis, namely, that it is the number of trials leading to the event that is regarded as uniformly distributed and not its probability. Many authors criticized Laplace’s hypothesis (Edgeworth 1885a; Fisher 1922a, 1956), arguing that other monotonic distributions of p, for example \( {\scriptscriptstyle \frac{1}{2}}Arc\mathrm{cos}(1-2p)\), could be equally suitable and yield different results. We shall discuss the hypothesis in greater detail at the end of the chapter.
- 18.
In his 1937 article, de Finetti used the term ‘equivalent events’, but the designation was not kept in later publications dealing with the concept. Today, the term ‘exchangeable events’ is used in all of the literature, even though some authors do not regard it as perfect.
- 19.
- 20.
Actually, de Finetti noted this event \( \frac{{E}^{\prime }}{{E}^{″}}\), but we shall use standard probability notations here.
- 21.
Allais won the Nobel Prize for Economics in 1988 for his contributions to market theory.
- 22.
The notion of semi-order rests on the idea that one alternative is preferable to another only if the utility of the first alternative exceeds the utility of the second by a certain constant threshold.
- 23.
Smets (1990) formulated these axioms slightly differently, calling the belief function bel:
-
1.
compositionality axiom: \( be{l}_{12}(A)\) is a function of A, \( be{l}_{1}and\ be{l}_{2}\) only.
-
2.
symmetry: \( be{l}_{1}\oplus \ be{l}_{2}=be{l}_{2}\oplus \ be{l}_{1}\).
-
3.
associativity: \( \left(be{l}_{1}\oplus \ be{l}_{2}\right)\oplus \ be{l}_{3}=be{l}_{1}\oplus \ (be{l}_{2}\oplus \ be{l}_{3})\).
-
4.
conditioning: if \( be{l}_{2}\) is such that \( {m}_{2}(B)=1\), then
$$\begin{array}{l}{m}_{12}(A)={\displaystyle \sum _{C\to \overline{B}}{m}_{1}(A\cup C)}for\ all\ A\to B\\ =0\ otherwise\end{array}$$ -
5.
internal symmetry : the mass given by \( {m}_{12}\) to \( A\in W\) is independent of the masses given by \( {m}_{1}\) (and \( {m}_{2}\) to propositions \( B\to \overline{A}\).
-
6.
auto functionality: \( \forall A\in \Omega,\ A\ne {1}_{\Omega },\text{m}_{12}(A)\) does not depend on \( {m}_{1}(X)\) for all \( X\to \overline{A}\).
-
7.
three elements: there are at least three elementary propositions in \(D\).
-
8.
continuity: let \( {m}_{2}(A)=1-\epsilon,\text{m}_{2}\left({1}_{\Omega }\right)=\epsilon \). Let \( {m}_{A}(A)=1\). For any \( be{l}_{1}\) defined on Ω, let \( be{l}_{1A}=be{l}_{1}\oplus \ be{l}_{A}\) then for all \( X\in \Omega \), \( \underset{\epsilon \to 0}{\mathrm{lim}}{m}_{12}(X)={m}_{1A}(X)\).
-
1.
- 24.
We have taken the objectivist model here, as the population observed is large. However, as noted earlier, the results with the subjectivist model would be very similar.
- 25.
Apart from Allais’s criticisms discussed above, we could, of course, have quoted many examples in economics making extensive use of subjective probability. Some of these examples are reported in the next section on issues raised by subjective analysis.
- 26.
Kahneman won the Nobel Prize for Economics in 2002 for this theory.
References
Allais, M. (1953). Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’école américaine. Econometrica, 21(4), 503–546.
Andersen, P. K., Borgan, Ø., Gill, R. D., & Keiding, N. (1993). Statistical models based on counting processes. New York/Berlin/Heidelberg: Springer.
Arnauld, A., & Nicole, P. (1662). La logique ou l’Art de penser. Paris: Chez Charles Savreux.
Ayton, P. (1997). How to be incoherent and seductive: Bookmaker’s odds and support theory. Organizational Behavior and Human Decision Processes, 72, 99–115.
Bayes, T. R. (1763). An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society of London, 53, 370–418.
Bernardo, J. M., & Smith, A. F. M. (1994). Bayesian theory. Chichester: Wiley.
Bernoulli, J. I. (1713). Ars conjectandi. Bâle: Impensis Thurnisiorum fratrum.
Bernoulli, D. (1738). Specimen theoriae novae de mensura sortis. Commentarii Academiae Scientiarum Imperialis Petropolitanae, V, 175–192.
Bernoulli, J.I., & Leibniz, G.W. (1692–1704). Lettres échangées. (French translation by Meusnier N. (2006). Quelques échanges ? Journal électronique d’Histoire des Probabilités et de la Statistique, 2(1), 1–12).
Bienaymé, J. (1838). Mémoire sur la probabilité des résultats moyens des observations; démonstration directe de la règle de Laplace. Mémoires Présentés à l’Académie Royale des Sciences de l’Institut de France, 5, 513–558.
Boole, G. (1854). An investigation of the laws of thought: On which are founded the mathematical theories of logic and probability. London: Walton and Maberly.
Browne, W. J. (1998). Applying MCMC methods to multi-level models. PhD thesis, University of Bath, Bath citeseerx.ist.edu: http://www.ams.ucsc.edu/~draper/browne-PhD-dissert. Accessed July 10, 2011.
Choquet, G. (1953). Theory of capacities. Annales de l’Institut Fourier, 5, 131–296.
Chung, K.-L. (1942). On mutually favourable events. The Annals of Statistics, 13, 338–349.
Cohen, M. R., & Nagel, E. (1934). An introduction to logic and scientific method. NewYork: Harcourt Brace.
Condorcet. (1778). Sur les probabilités. Histoire de l’Académie Royale de Sciences, 1781, 43–46.
Courgeau, D. (2002). Evolution ou révolutions dans la pensée démographique? Mathématiques et Sciences Humaines, 40(160), 49–76.
Courgeau, D. (2004b). Probabilités, démographie et sciences sociales. Mathématiques et Sciences Humaines, 42(3), 5–19.
Courgeau, D. (2007a). Multilevel synthesis. From the group to the individual. Dordrecht: Springer.
Courgeau, D. (2007b). Inférence statistique, échangeabilité et approche multiniveau. Mathématiques et Sciences Humaines, 45(179), 5–19.
Courgeau, D., & Lelièvre, E. (1989). Analyse démographique des biographies. Paris: Ined (English translation: (1992). Event history analysis in demography. Oxford: Clarendon Press. Spanish translation: (2001). Análisis demográfico de las biografías. México: El Colegio de México).
Cox, R. (1961). The algebra of probable inference. Baltimore: The John Hopkins Press.
Cox, D. R., & Oakes, D. (1984). Analysis of survival data. London/New York: Chapman and Hall.
de Finetti, B. (1931a). Sul significato soggettivo della probabilità. Fundamenta Mathematicae, 17, 298–329.
de Finetti, B. (1937). La prévision: ses lois logiques, ses sources subjectives. Annales de l’Institut Henri Poincaré, 7, 1–68 (Paris).
de Finetti, B. (1951). Recent suggestions for the reconciliation of theories of probability. In J. Neyman (Ed.), Proceedings of the second Berkeley symposium on mathematical statistics and probability (pp. 217–225). Berkeley: University of California Press.
de Finetti, B. (1952). Sulla preferibilità. Giornale degli Economisti e Annali de Economia, 11, 685–709.
de Montmort, P. R. (1713). Essay d’analyse sur les jeux de hazard (2nd ed.). Paris: Jacques Quillau.
DeGroot, M. H. (1970). Optimal statistical decision. New York: McGraw-Hill.
Dempster, A. P. (1967). Upper and lower probabilities induced by a multilevel mapping. Annals of Mathematical Statistics, 38, 325–339.
Dempster, A. P. (1968). A generalization of Bayesian inference. Journal of the Royal Statistical Society, Series B, 30, 205–245.
Dubois, D., & Prade, H. (1988). An introduction to possibilistic and fuzzy logics. In P. Smets, A. Mandani, D. Dubois, & H. Prade (Eds.), Non standard logics for automated reasoning (pp. 287–326). New York: Academic Press.
Ellis, R. L. (1849). On the Foundations of the theory of probability. Transactions of the Cambridge Philosophical Society, VIII, 1–16.
Eriksson, L., & Hájek, A. (2007). What are degrees of belief? Studia Logica, 86, 185–215.
Fishburn, P. C. (1964). Decision and value theory. New York: Wiley.
Fishburn, P. C. (1975). A theory of subjective probability and expected utilities. Theory and Decision, 6, 287–310.
Fishburn, P. C. (1986). The axioms of subjective probability. Statistical Science, 1(3), 335–345.
Fisher, R. A. (1922a). The mathematical theory of probability. London: Macmillan.
Fisher, R. A. (1956). Statistical methods and scientific inference. Edinburgh: Oliver and Boyd.
Freund, J. E. (1965). Puzzle or paradox? The American Statistician, 19(4), 29–44.
Frischhoff, B., Slovic, B., & Lichteinstein, S. (1978). Fault trees: Sensitivity of estimated failure probabilities to problem representation. Journal of Experimental Psychology. Human Perception and Performance, 4, 330–344.
Gacôgne, L. (1993). About a foundation of the Dempster’s rule. (Rapport 93/27). Laforia.
Gärdenfors, P., Hansson, B., & Sahlin, N.-E. (Eds.). (1983). Evidential value: Philosophical, judicial and psychological aspects of a theory. Lund: Gleerups.
Gelman, A., Karlin, J. B., Stern, H. S., & Rubin, D. B. (1995). Bayesian data analysis. New York: Chapman and Hall.
Gill, J. (2008). Bayesian methods. A social and behavioral sciences approach. Boca Raton: Chapman & Hall.
Gillies, D. (2000). Philosophical theories of probability. London/New York: Routledge.
Goldstein, H. (2003). Multilevel statistical models. London: Edward Arnold.
Good, I. J. (1962). Subjective probability as the measure of a non-measurable set. In E. Nagel, P. Suppes, & A. Tarski (Eds.), Logic, methodology and philosophy of science (pp. 319–329). Stanford: Stanford University Press.
Grether, D. M., & Plott, C. R. (1979). Economic theory of choice and the preference reversal phenomenon. The American Economic Review, 69(4), 623–638.
Hacking, I. (1965). Logic of statistical inference. Cambridge: Cambridge University Press.
Hájek, A. (2008a). Dutch book arguments. In P. Anand, P. Pattanaik, & C. Pup (Eds.), The Oxford handbook of corporate social responsibility, Phil papers: http://philrsss.anu.edu.au/people-defaults/alanh/papers/DBA.pdf. Accessed July 10, 2011.
Hájek, A. (2008b). Probability. In B. Gold (Ed.), Current issues in the philosophy of mathematics: From the perspective of mathematicians, Mathematical Association of America, Phil papers. http://philrsss.anu.edu.au/people-defaults/alanh/papers/overview.pdf. Accessed July 10, 2011.
Hooper, G. (1699). A calculation of the credibility of human testimony. Anonymous translation in Philosophical Transactions of the Royal Society, 21, 359–365.
Ibrahim, J. G., Chen, M.-H., & Sinha, D. (2001). Bayesian survival analysis. New York/Berlin/Heidelberg: Springer.
Jaynes, E. T. (2003). Probability theory: The logic of science. Cambridge: Cambridge University Press.
Jeffreys, H. (1939). Theory of probability. New York: Clarendon Press.
Johnson, W. E. (1932). Probability: The deductive and inductive problem. Mind, 41, 409–423.
Kahneman, D., & Tversky, A. (1972). Subjective probability: A judgement of representativeness. Cognitive Psychology, 3, 430–454.
Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263–291.
Kalbfleisch, J. D., & Prentice, R. L. (1980). The statistical analysis of failure time data. New York/Chichester/Brisbane/Toronto: Wiley.
Kaplan, M. (1996). Decision theory as philosophy. Cambridge: Cambridge University Press.
Keynes, J. M. (1921). A treatise on probability. London: Macmillan.
Kolmogorov, A. (1933). Grundbegriffe der wahrscheinlichkeitsrenung. In Ergebisne der Mathematik, (Vol. 2) Berlin: Springer (English translation, Morrison, N. (1950). Foundations of the theory of probability New York: Chelsea).
Kraft, C. H., Pratt, J. W., & Seidenberg, A. (1959). Intuitive probability on finite sets. Annals of Mathematical Statistics, 30, 408–419.
Kyburg, H. (1978). Subjective probability: Criticisms, reflections and problems. Journal of Philosophical Logic, 7, 157–180.
Lambert, J. H. (1764). Neues Organon oder gedanken über die Erforschung und Bezeichnung des wahren und dessen underscheidung von irrthum und schein. Leipzig: Johan Wendler.
Laplace, P. S. (1774). Mémoire sur la probabilité des causes par les événements. Mémoires de l’Académie Royale des Sciences de Paris, Tome VI, 621–656.
Laplace, P. S. (1783a). Mémoire sur les approximations des formules qui sont fonctions de très-grands nombres (suite). Mémoires de l’Académie Royale des sciences de Paris, 1786, 423–467.
Laplace, P. S. (1812). Théorie analytique des Probabilités (Vols. 2). Paris: Courcier Imprimeur.
Laplace, P. S. (1814). Essai philosophique sur les probabilités. Paris: Courcier Imprimeur.
Lee, P. M. (1989). Bayesian statistics. London: Arnold.
Leonard, T., & Hsu, S. J. (1999). Bayesian methods. An analysis for statisticians and interdisciplinary researchers. Cambridge: Cambridge University Press.
Lindley, D. V., & Novick, M. R. (1981). The role of exchangeability in inference. The Annals of Statistics, 9, 45–58.
Loomes, G., & Sugden, R. (1982). Regret theory: An alternative theory of rational choice under uncertainty. The Economic Journal, 92, 805–824.
Luce, R. D., & Krantz, D. H. (1971). Conditional expected utility. Econometrica, 39(2), 253–271.
Luce, R. D., & Suppes, P. (1965). Preference, utility and subjective probability. In R. D. Luce, R. R. Bush, & E. H. Galanter (Eds.), Handbook of mathematical psychology (Vol. 3, pp. 249–410). New York: Wiley.
Machina, M. J. (1982). “Expected utility” analysis without the independence axiom. Econometrica, 50(2), 277–323.
Mandel, D. R. (2005). Are risk assessments of a terrorist attack coherent? Journal of Experimental Psychology Applied, 11, 277–288.
Mandel, D. R. (2008). Violations of coherence in subjective probability: A representational and assessment processes account. Cognition, 106, 130–156.
Matalon, B. (1967). Epistémologie des probabilités. In J. Piaget (Ed.), Logique et connaissance scientifique (pp. 526–553). Paris: Gallimard.
McCrimmon, K., & Larson, S. (1979). Utility theory: Axioms versus “paradoxes”. In M. Allais & O. Hagen (Eds.), Expected utility hypotheses and the Allais paradox (pp. 333–409). Dordrecht: D. Reidel.
Morrison, D. (1967). On the consistency of preferences in Allais’s paradox. Behavioral Science, 12, 373–383.
Muliere, P., & Parmigiani, G. (1993). Utility and means in the 1930s. Statistical Science, 8(4), 421–432.
Narens, L. (1976). Utility, uncertainty and trade-off structures. Journal of Mathematical Psychology, 13, 296–332.
Pascal, B. (1670). Pensées. Paris: Édition de Port-Royal.
Pearson, K. (1920). The fundamental problem of practical statistics. Biometrika, 13(1), 1–16.
Pearson, K. (1925). Bayes’ theorem, examined in the light of experimental sampling. Biometrika, 17(3/4), 388–442.
Piaget, J. (1967). Les deux problèmes principaux de l’épistémologie des sciences de l’homme. In J. Piaget (Ed.), Logique et connaissance Scientifique (pp. 1114–1146). Paris: Gallimard.
Poisson, S.-D. (1837). Recherches sur la probabilité des jugements en matière criminelle et en matière civile. Paris: Bachelier.
Preston, M. G., & Baratta, P. (1948). An experimental study of the auction value of an uncertain outcome. The American Journal of Psychology, 61, 183–193.
Ramsey, F. P. (1926). Truth and probability. In Ramsey, F.P. (1931), The foundations of mathematics and other logical essays, R.B. Braithwaite ed, Ch. VII, pp. 156–198. London: Kegan, Trubner & Co., New York: Harcourt, Brace and Company.
Ramsey, F. P. (1931). In R. B. Braithwaite (Ed.), The foundations of mathematics and other logical essays. London/New York: Kegan, Trubner & Co/Harcourt, Brace and Company.
Robert, C. P. (2006). Le choix bayésien. Paris: Springer-Verlag France.
Roberts, H. V. (1974). Reporting of Bayesian studies. In S. E. Fienberg & A. Zellner (Eds.), Studies in Bayesian econometrics and statistics: In honor of Leonard J. Savage (pp. 465–483). Amsterdam: North Holland.
Rouanet, H., Bernard, J.-M., Bert, M.-C., Lecoutre, B., Lecoutre, M.-P., & Le Roux, B. (1998). New ways in statistical methodology. From significance tests to Bayesian inference. Bern: Peter Lang.
Rouanet, H., Lebaron, F., Le Hay, V., Ackermann, W., & Le Roux, B. (2002). Régression et analyse géométrique des données: réflexions et suggestions. Mathématiques et Sciences Humaines, 160, 13–45.
Savage, L. J. (1954). The foundations of statistics. New York: Wiley.
Scott, D. (1964). Measurement structures and linear inequalities. Journal of Mathematical Psychology, 1, 233–247.
Seidenfeld, T., Schervish, M. J., & Kadane, J. B. (1990). When fair betting odds are not degrees of belief. Philosophical Science Association, 1, 517–524.
Shafer, G. (1976). A mathematical theory of evidence. Princeton: Princeton University Press.
Shafer, G. (1979). Allocations of probability. Annals of Probability, 7(5), 827–839.
Shafer, G. (1985). Conditional probability. International Statistical Review, 53(3), 261–277.
Shafer, G. (1986). Savage revisited. Statistical Science, 1(4), 463–501.
Shafer, G. (2001). The notion of event in probability and causality. Situating myself relative to Bruno de Finetti. Unpublished paper, presented in Pisa and in Bologna march 2001, 14 p.
Shafer, G., Gilett, P. R., & Scherl, R. B. (2000). The logic of events. Annals of Mathematics and Artificial Intelligence, 28, 315–389.
Simpson, E. H. (1951). The interpretation of interaction in contingency tables. Journal of the Royal Statistical Society Series B, 13, 238–241.
Smets, P. (1988). Belief functions. In P. Smets, A. Mandani, P. Dubois, & H. Prade (Eds.), Non standard logics for automated reasoning (pp. 253–286). London: Academic Press.
Smets, P. (1990). Constructing pignistic probability function in a context of uncertainty. In M. Henrion, R. D. Shachter, L. N. Kanal, & J. F. Lemmer (Eds.), Uncertainty in artificial intelligence (Vol. 5, pp. 29–40). Amsterdam: North Holland.
Smets, P. (1991). Probability of provability and belief functions. Logique et Analyse, 133–134, 177–195.
Smets, P. (1994). What is Dempster-Shafer’s model? In R. R. Yager, M. Fedrizzi, & J. Kacprzyk (Eds.), Advances in the Dempster-Shafer theory of evidence (pp. 5–34). New York/Chichester/Brisbane/Toronto: Wiley.
Smets, P. (1997). The normative representation of quantified beliefs by belief functions. Artificial Intelligence, 92, 229–242.
Smets, P. (1998). The transferable belief model for quantified belief representation. In P. Smets (Ed.), Handbook of defeasible reasoning and uncertainty management systems, Vol. 1: Quantified representation of uncertainty & imprecision (pp. 267–301). Dordrecht: Kluwer.
Smith, C. A. B. (1961). Consistency in statistical inference and decision, (with discussion). Journal of the Royal Statistical Society Series B, 23, 1–25.
Smith, C. A. B. (1965). Personal probability and statistical analysis, (with discussion). Journal of the Royal Statistical Society Series A, 128, 469–499.
Starmer, C. (1992). Testing new theories of choice under uncertainty using the common consequence effect. Review of Economic Studies, 59(4), 813–830.
Starmer, C. (2000). Developments in non expected-utility theory: The hunt for a descriptive theory of choice under risk. Journal of Economic Literature, 38(2), 332–382.
Stigler, S. M. (1986). The history of statistics: the measurement of uncertainty before 1900. Cambridge: Belknap Press of Harvard University Press.
Stigum, B. P. (1972). Finite state space and expected utility maximization. Econometrica, 40, 253–259.
Suppes, P. (1956). The role of subjective probability and utility in decision-making. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955(5), 61–73.
Suppes, P. (1960). Some open problems in the foundations of subjective probability. In R. E. Machol (Ed.), Information and decision processes (pp. 129–143). New York: McGraw-Hill.
Suppes, P. (1974). The measurement of belief. Journal of the Royal Statistical Society Series B (Methodological), 36(2), 160–191.
Suppes, P. (1976). Testing theories and the foundations of statistics. In W. L. Harper & C. A. Hooker (Eds.), Foundations of probability theory, statistical inference, and statistical theories of science (Vol. II, pp. 437–457). Dordrecht: Reidel.
Suppes, P. (2002a). Representation and invariance of scientific structures. Stanford: CSLI Publications.
Suppes, P., & Zanotti, M. (1975). Necessary and sufficient conditions for existence of a unique measure strictly agreeing with a qualitative probability ordering. Journal of Philosophical Logic, 5, 431–438.
Suppes, P., & Zanotti, M. (1982). Necessary and sufficient qualitative axioms for conditional probability. Zeitschrift für Wahrscheinlichkeitstheorie und Werwande Gebiete, 60, 163–169.
Tversky, A. (1974). Assessing uncertainty. Journal of the Royal Statistical Society Series B (Methodological), 36(2), 148–159.
Tversky, A., & Kahneman, D. (1981). The framing of decisions and the psychology of choice. Science, 211(30), 453–457.
Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5, 297–323.
Tversky, A., & Koeler, D. J. (1994). Support theory: A nonextensional representation of subjective probability. Psychological Review, 101, 547–567.
Venn, J. (1866). The logic of chance. London: Macmillan.
von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behaviour. Princeton: Princeton University Press.
Weber, M. (1998). The resilience of the Allais paradox. Ethics, 109(1), 94–118.
Weber, B. (2007). The effects of losses and event splitting on the Allais paradox. Judgment and Decision Making, 2, 115–125.
Yager, R. R., & Liu, L. (Eds.). (2007). Classic works on the Dempster-Shafer theory of belief functions. Heidelberg: Springer.
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.
Zadeh, L. A. (1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1, 3–28.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Courgeau, D. (2012). The Epistemic Approach: Subjectivist Interpretation. In: Probability and Social Science. Methodos Series, vol 10. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2879-0_2
Download citation
DOI: https://doi.org/10.1007/978-94-007-2879-0_2
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-2878-3
Online ISBN: 978-94-007-2879-0
eBook Packages: Humanities, Social Sciences and LawSocial Sciences (R0)