##
**Comparative probability for conditional events: A new look through coherence.**
*(English)*
Zbl 0785.90006

Summary: We study, from the standpoint of coherence, comparative probabilities on an arbitrary family \(\mathcal E\) of conditional events. Given a binary relation \(\prec\cdot\), coherence conditions on \(\prec\cdot\) are related to de Finetti’s coherent betting system: we consider their connections to the usual properties of comparative probability and to the possibility of numerical representations of \(\prec\cdot\). In this context, the numerical reference frame is that of de Finetti’s coherent subjective conditional probability, which is not introduced (as in Kolmogoroff’s approach) through a ratio between probability measures.

Another relevant feature of our approach is that the family \(\mathcal E\) need not have any particular algebraic structure, so that the ordering can be initially given for a few conditional events of interest and then possibly extended by a step-by-step procedure, preserving coherence.

Another relevant feature of our approach is that the family \(\mathcal E\) need not have any particular algebraic structure, so that the ordering can be initially given for a few conditional events of interest and then possibly extended by a step-by-step procedure, preserving coherence.

PDF
BibTeX
XML
Cite

\textit{G. Coletti} et al., Theory Decis. 35, No. 3, 237--258 (1993; Zbl 0785.90006)

Full Text:
DOI

### References:

[1] | Coletti, G.: 1990, ?Coherent qualitative probability?,Journal of Mathematical Psychology 34, 297-310. · Zbl 0713.60003 |

[2] | Coletti, G.: 1993, ?Comparative probabilities ruled by coherence conditions and its use in expert systems?,International Journal of General Systems (to appear). · Zbl 0797.60003 |

[3] | Coletti, G., Gilio, A. and Scozzafava, R.: 1990, ?Coherent qualitative probability and uncertainty in Artificial Intelligence?,Proc. 8th International Congress of Cybernetics and Systems (Vol. I, Ed. C.N. Manikopoulos), NJIT Press, New York, pp. 132-138. |

[4] | Coletti, G., Gilio, A. and Scozzafava, R: 1991a, ?Conditional events with vague information in expert systems?,Lecture Notes in Computer Science (Eds. B. Bouchon-Meunier, R.R. Yager and L.A. Zadeh), No. 521, pp. 106-114. · Zbl 0800.68922 |

[5] | Coletti, G., Gilio, A. and Scozzafava, R.: 1991b, ?Assessment of qualitative judgments for conditional events in expert systems?,Lecture Notes in Computer Science (Eds. R. Kruse and P. Siegel), No. 548, pp. 135-140. · Zbl 0800.68922 |

[6] | Coletti, G. and Scozzafava, R.: 1993, ?A coherent qualitative Bayes’ theorem and its application in artificial intelligence?,2nd Int. Symp. on Uncertainty Modeling and Analysis, Maryland (Ed. B.M. Ayyub), pp. 40-44. |

[7] | De Finetti, B.: 1949, ?Sull impostazione assiomatica del calcolo delle probabilità?,Annali Univ. Trieste,19, 3-55 (English transl. (1972): Ch. 5 inProbability, Induction, Statistics, London: Wiley). |

[8] | De Finetti, B.: 1970,Teoria delle probabilità, G. Einaudi, Torino (English transl. (1974):Theory of Probability, Vols. 1 and 2, Wiley & Sons, Chichester). |

[9] | Fenchel, W.: 1951, ?Convex cones, sets and functions?,Lecture at Princeton University, Spring term. · Zbl 0053.12203 |

[10] | Gale, D.: 1960,The Theory of Linear Economic Models, McGraw-Hill, New York. · Zbl 0114.12203 |

[11] | Gilio, A. and Scozzafava, R.: 1988, ?Le probabilità condizionate coerenti nei sistemi esperti?,Atti Giornate AIRO su Ricerca Operativa e Intelligenza Artificiale, Centro di Ricerca IBM, Pisa, pp. 317-330. |

[12] | Goodman, I.R. and Nguyen, H.T.: 1988, ?Conditional objects and the modeling of uncertainties?, inFuzzy Computing (Eds. M. Gupta and T. Yamakawa), North Holland, Amsterdam, pp. 119-138. |

[13] | Holzer, S.: 1985, ?On coherence and conditional prevision?,Boll. Un. Mat. Ital. 4(6), 441-460. · Zbl 0584.60001 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.