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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Order-distance and other metric-like functions on jointly distributed random variables


Authors: Ehtibar N. Dzhafarov and Janne V. Kujala
Journal: Proc. Amer. Math. Soc. 141 (2013), 3291-3301
MSC (2010): Primary 60B99; Secondary 81Q99, 91E45
Published electronically: May 15, 2013
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Abstract: We construct a class of real-valued nonnegative binary functions on a set of jointly distributed random variables. These functions satisfy the triangle inequality and vanish at identical arguments (pseudo-quasi-metrics). We apply these functions to the problem of selective probabilistic causality encountered in behavioral sciences and in quantum physics. The problem reduces to that of ascertaining the existence of a joint distribution for a set of variables with known distributions of certain subsets of this set. Any violation of the triangle inequality by one of our functions when applied to such a set rules out the existence of the joint distribution. We focus on an especially versatile and widely applicable class of pseudo-quasi-metrics called order-distances. We show, in particular, that the Bell-CHSH-Fine inequalities of quantum physics follow from the triangle inequalities for appropriately defined order-distances.


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Additional Information

Ehtibar N. Dzhafarov
Affiliation: Department of Psychological Sciences, Purdue University, West Lafayette, Indiana 47907
Email: ehtibar@purdue.edu

Janne V. Kujala
Affiliation: Department of Mathematics, University of Jyväskylä, Jyväskylä, Finland
Email: jvk@iki.fi

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11575-3
PII: S 0002-9939(2013)11575-3
Keywords: Bell-CHSH-Fine inequalities, Einstein-Podolsky-Rosen paradigm, probabilistic causality, pseudo-quasi-metrics on random variables, quantum entanglement, selective influences
Received by editor(s): October 26, 2011
Received by editor(s) in revised form: November 18, 2011
Published electronically: May 15, 2013
Additional Notes: The first author’s work was supported by AFOSR grant FA9550-09-1-0252
The second author’s work was supported by Academy of Finland grant 121855
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.