This paper studies graphoid properties for epistemic irrelevance in sets of desirable gambles. For that aim, the basic operations of conditioning and marginalization are expressed in terms of variables. Then, it is shown that epistemic irrelevance is an asymmetric graphoid. The intersection property is verified in probability theory when the global probability distribution is positive in all the values. Here it is always verified due to the handling of zero probabilities in sets of gambles. An asymmetrical D-separation principle is also presented, by which this type of independence relationships can be represented in directed acyclic graphs.
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References
I. Couso, S. Moral and P. Walley, A survey of concepts of independence for imprecise probabilities, Risk, Decision and Policy 5 (2000) 165–181.
F. Cozman, Credal networks, Artificial Intelligence 120 (2000) 199–233.
F. Cozman and P. Walley, Graphoid properties of epistemic irrelevance and independence, in: Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications (ISIPTA ’01), eds. G. de Cooman, T. Fine and T. Seidenfeld (Shaker Publishing, 2001) pp. 112–121.
A. Dawid, Conditional independence, in: Encyclopedia of Statistical Sciences, Update Vol. 2, eds. S. Kotz, C.B. Read and D.L. Banks (Wiley, New York, 1999) pp. 146–153.
S. Moral, Epistemic irrelevance on sets of desirable gambles, in: Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications, eds. G. de Cooman, T. Fine, and T. Seidenfeld (Shaker Publishing, 2001) pp. 247–254.
S. Moral and A. Cano, Strong Conditional Independence for Credal Sets, Annals of Mathematics and Artificial Intelligence 35 (2002) 295–321.
S. Moral and N. Wilson, Revision rules for convex sets of probabilities, in: Mathematical Models for Handling Partial Knowledge in Artificial Intelligence, eds. G. Coletti, D. Dubois and R. Scozzafava (Plenum Press, 1995) pp. 113–128.
J. Pearl, Probabilistic Reasoning with Intelligent Systems (Morgan & Kaufman, San Mateo, 1988).
J. Pearl and T. Verma, The logic of representing dependencies by directed graphs, in: Proceedings of the AAAI'87 Conference (AAAI/MIT Press, 1987) pp. 347–379.
G. Shafer and P. Shenoy, Local Computation in Hypertrees, Working Paper N. 201, School of Business, University of Kansas, Lawrence, 1988.
M. Studeny, On separation criterion and recovery algorithm for chain graphs, in: Proceedings of the Twelfth Annual Conference on Uncertainty in Artificial Intelligence (UAI–96) (Portland, Oregon, 1996) pp. 509–516.
M. Studeny, Semigraphoids and structures of probabilistic conditional independence, Annals of Mathematics and Artificial Intelligence 21 (1997) 71–98.
B. Vantaggi, Graphical models for conditional independence structures, in: Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications, eds. G. de Cooman, T. Fine and T. Seidenfeld (Shaker Publishing, 2001) pp. 332–341.
P. Vicig, Epistemic independence for imprecise probabilities, International Journal of Approximate Reasoning 24 (2000) 235–250.
P. Walley, Statistical Reasoning with Imprecise Probabilities (Chapman and Hall, London, 1991).
P. Walley, Towards a unified theory of imprecise probability, International Journal of Approximate Reasoning 24 (2000) 125–148.
N. Wilson and S. Moral, A logical view of probability, in: Proceedings of the Eleventh European Conference on Artificial Intelligence (ECAI'94), ed. A. Cohn (Wiley, London, 1994) pp. 386–390.
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Moral, S. Epistemic irrelevance on sets of desirable gambles. Ann Math Artif Intell 45, 197–214 (2005). https://doi.org/10.1007/s10472-005-9011-0
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DOI: https://doi.org/10.1007/s10472-005-9011-0