Abstract
In this paper we consider, in the framework of coherence, four different definitions of conjunction among conditional events. In each of these definitions the conjunction is still a conditional event. We first recall the different definitions of conjunction; then, given a coherent probability assessment (x, y) on a family of two conditional events \(\{A|H,B|K\}\), for each conjunction \((A|H) \wedge (B|K)\) we determine the (best) lower and upper bounds for the extension \(z=P[(A|H) \wedge (B|K)]\). We show that, in general, these lower and upper bounds differ from the classical Fréchet-Hoeffding bounds. Moreover, we recall a notion of conjunction studied in recent papers, such that the result of conjunction of two conditional events A|H and B|K is (not a conditional event, but) a suitable conditional random quantity, with values in the interval [0, 1]. Then, we remark that for this conjunction, among other properties, the Fréchet-Hoeffding bounds are preserved.
G. Sanfilippo was partially supported by the National Group for Mathematical Analysis, Probability and their Applications (GNAMPA – INdAM).
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References
Adams, E.W.: The Logic of Conditionals. Reidel, Dordrecht (1975)
Benferhat, S., Dubois, D., Prade, H.: Nonmonotonic reasoning, conditional objects and possibility theory. Artif. Intell. 92, 259–276 (1997)
Bochvar, D., Bergmann, M.: On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus. Hist. Philos. Log. 2(1–2), 87–112 (1981)
Ciucci, D., Dubois, D.: Relationships between connectives in three-valued logics. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) IPMU 2012. CCIS, vol. 297, pp. 633–642. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31709-5_64
Ciucci, D., Dubois, D.: A map of dependencies among three-valued logics. Inf. Sci. 250, 162–177 (2013)
Coletti, G., Gervasi, O., Tasso, S., Vantaggi, B.: Generalized bayesian inference in a fuzzy context: from theory to a virtual reality application. Comput. Stat. Data Anal. 56(4), 967–980 (2012)
Coletti, G., Scozzafava, R.: Conditional probability, fuzzy sets, and possibility: A unifying view. Fuzzy Sets Syst. 144, 227–249 (2004)
Coletti, G., Scozzafava, R., Vantaggi, B.: Coherent conditional probability, fuzzy inclusion and default rules. In: Yager, R., Abbasov, A., Reformat, M., Shahbazova, S. (eds.) Soft Computing: State of the Art Theory and Novel Applications. Studies in Fuzziness and Soft Computing, vol. 291, pp. 193–208. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-34922-5_14
Coletti, G., Scozzafava, R., Vantaggi, B.: Possibilistic and probabilistic logic under coherence: Default reasoning and System P. Math. Slovaca 65(4), 863–890 (2015)
Dubois, D., Prade, H.: Conditional objects as nonmonotonic consequence relationships. IEEE Trans. Syst. Man Cybern. 24(12), 1724–1740 (1994)
de Finetti, B.: La Logique de la Probabilité. In: Actes du Congrès International de Philosophie Scientifique, Paris, 1935, pp. IV 1–IV 9. Hermann et C.ie, Paris (1936)
de Finetti, B.: Teoria delle probabilità. In: Einaudi (ed.) 2 vol., Torino (1970). English version: Theory of Probability 1 (2). Chichester, Wiley, 1974 (1975)
Gale, D.: The Theory of Linear Economic Models. McGraw-Hill, New York (1960)
Gilio, A.: Criterio di penalizzazione e condizioni di coerenza nella valutazione soggettiva della probabilità. Boll. Un. Mat. Ital. 4–B(3, Serie 7), 645–660 (1990)
Gilio, A.: \(C_0\)-Coherence and extension of conditional probabilities. In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (eds.) Bayesian Statistics 4, pp. 633–640. Oxford University Press (1992)
Gilio, A.: Probabilistic consistency of knowledge bases in inference systems. In: Clarke, M., Kruse, R., Moral, S. (eds.) ECSQARU 1993. LNCS, vol. 747, pp. 160–167. Springer, Heidelberg (1993). https://doi.org/10.1007/BFb0028196
Gilio, A.: Algorithms for conditional probability assessments. In: Berry, D.A., Chaloner, K.M., Geweke, J.K. (eds.) Bayesian Analysis in Statistics and Econometrics: Essays in Honor of Arnold Zellner, pp. 29–39. Wiley, New York (1996)
Gilio, A.: Generalizing inference rules in a coherence-based probabilistic default reasoning. Int. J. Approx. Reason. 53(3), 413–434 (2012)
Gilio, A., Sanfilippo, G.: Quasi conjunction and p-entailment in nonmonotonic reasoning. In: Borgelt, C. (ed.) Combining Soft Computing and Statistical Methods in Data Analysis. AISC, vol. 77, pp. 321–328. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14746-3_40
Gilio, A., Sanfilippo, G.: Coherent conditional probabilities and proper scoring rules. In: Proceedings of ISIPTA 2011, Innsbruck, pp. 189–198 (2011)
Gilio, A., Sanfilippo, G.: Quasi conjunction, quasi disjunction, t-norms and t-conorms: probabilistic aspects. Inf. Sci. 245, 146–167 (2013)
Gilio, A., Sanfilippo, G.: Conditional random quantities and compounds of conditionals. Stud. Log. 102(4), 709–729 (2014)
Gilio, A., Sanfilippo, G.: Conjunction and disjunction among conditional events. In: Benferhat, S., Tabia, K., Ali, M. (eds.) IEA/AIE 2017. LNCS (LNAI), vol. 10351, pp. 85–96. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-60045-1_11
Gilio, A., Sanfilippo, G.: Generalized logical operations among conditional events. Applied Intelligence (in press). https://doi.org/10.1007/s10489-018-1229-8
Goodman, I.R., Nguyen, H.T.: Conditional objects and the modeling of uncertainties. In: Gupta, M.M., Yamakawa, T. (eds.) Fuzzy Computing, pp. 119–138. North-Holland, Amsterdam (1988)
Goodman, I.R., Nguyen, H.T., Walker, E.A.: Conditional Inference and Logic for Intelligent Systems: A Theory of Measure-Free Conditioning. North-Holland, Amsterdam (1991)
Milne, P.: Bruno de Finetti and the logic of conditional events. Br. J. Philos. Sci. 48(2), 195–232 (1997)
Pelessoni, R., Vicig, P.: The Goodman-Nguyen relation within imprecise probability theory. Int. J. Approx. Reason. 55(8), 1694–1707 (2014)
Sanfilippo, G., Pfeifer, N., Over, D.E., Gilio, A.: Probabilistic inferences from conjoined to iterated conditionals. Int. J. Approx. Reason. 93, 103–118 (2018)
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We thank Angelo Gilio and the three anonymous reviewers for their useful comments and suggestions.
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Sanfilippo, G. (2018). Lower and Upper Probability Bounds for Some Conjunctions of Two Conditional Events. In: Ciucci, D., Pasi, G., Vantaggi, B. (eds) Scalable Uncertainty Management. SUM 2018. Lecture Notes in Computer Science(), vol 11142. Springer, Cham. https://doi.org/10.1007/978-3-030-00461-3_18
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