Abstract
A definition of stochastic independence which avoids the inconsistencies (related to events of probability 0 or 1) of the classic one has been proposed by Coletti and Scozzafava for two events. We extend it to conditional independence among finite sets of events. In particular, the case of (finite) discrete random variables is studied. We check which of the relevant properties connected with graphical structures hold. Hence, an axiomatic characterization of these independence models is given and it is compared to the classic ones.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Billingsley, Probability and Measure (Wiley, New York, 1995).
G. Coletti, Coherent numerical and ordinal probabilistic assessments, IEEE Trans. Systems, Man, Cybernetics 24(12) (1994) 1747-1754.
G. Coletti, Updating and extension of coherent conditional probabilities, in: Proc. of the American Statist. Assoc. Conf., Section on Bayesian Statist. Science, Chicago (1996) pp. 36-41.
G. Coletti and R. Scozzafava, Characterization of coherent conditional probabilities as a tool for their assessment and extension, Internat. J. Uncertainty, Fuzziness and Knowledge-Based System 4(2) (1996) 103-127.
G. Coletti and R. Scozzafava, Null events and stochastical independence, Kibernetika 34(1) (1998) 69-78.
G. Coletti and R. Scozzafava, Zero probabilities in stochastic independence, in: Information, Uncertainty, Fusion, eds. B. Bouchon-Meunier, R.R. Yager and L.A. Zadeh (Kluwer, Dordrecht, 2000) pp. 185-196 (selected papers from IPMU ‘98).
G. Coletti and R. Scozzafava, Conditioning and inference in intelligent systems, Soft Computing 3 (1999) 118-130.
L. Crisma, The notion of stochastic independence in the theory of coherent probabilities, Technical Report 1/99, Dipartimento di Matematica applicata “B. de Finetti”, University of Trieste (1999).
A.P. Dawid, Conditional independence in statistical theory, J. Roy. Statist. Soc. B 41 (1979) 15-31.
F.T. de Dombal and F. Gremy (eds.), Decision Making and Medical Care (North-Holland, 1976).
B. de Finetti, Sull'impostazione assiomatica del calcolo delle probabilitá, Annali dell'Universitá di Trieste 19 (1949) 3-55. Eng. transl.: Ch. 5 in Probability, Induction, Statistics (Wiley, London, 1972). B. Vantaggi / Conditional independence in a coherent finite setting 313
L.E. Dubins, Finitely additive conditional probabilities, conglomerability and disintegration, Ann. Probab. 3 (1975) 89-99.
P. Hajek, T. Havranek and R. Jirousek, Uncertain Information Processing in Expert Systems (CRC Press, Boca Raton, FL, 1992).
J.R. Hill, Comment on “Graphical models”, Statist. Sci. 8 (1993) 258-261.
P.H. Krauss, Representation of conditional probability measures on Boolean algebras, Acta Math. Acad. Scient. Hungar 19 (1968) 229-241.
S.L. Lauritzen, Graphical Models (Clarendon Press, Oxford, 1996).
J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference (Morgan Kaufmann, San Mateo, CA, 1988).
A. Rényi, On conditional probability spaces generated by a dimensionally ordered set of measures, Theor. Probab. Appl. 1 (1956) 61-71.
R. Scozzafava, Subjective conditional probability and coherence principles for handling partial information, Mathware & Soft Computing 3 (1996) 183-192.
M. Studeny and R.R. Bouckaert, On chain graph models for description of conditional independence structures, Ann. Statist. 26 (4) (1998) 1434-1495.
B. Vantaggi, Conditional independence and graphical models in a coherent setting, PhD thesis, Università di Perugia, Italy (1999).
J. Witthaker, Graphical Models in Applied Multivariate Statistic (Wiley, New York, 1990).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Vantaggi, B. Conditional Independence in A Coherent Finite Setting. Annals of Mathematics and Artificial Intelligence 32, 287–313 (2001). https://doi.org/10.1023/A:1016730003879
Issue Date:
DOI: https://doi.org/10.1023/A:1016730003879