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Compatibility and statistical inference for data in measurement model of Foulis and Randall. (English) Zbl 0937.05075
This paper is motivated by the work of Foulis and Randall on operational statistics; see D. J. Foulis and C. H. Randall [J. Math. Phys. 13, 1667-1675 (1972; Zbl 0287.60002)]. Let \(G=(X,E)\) be a (possibly infinite) graph and let \(H=\bigl (X,{\mathcal E}(G)\bigr)\) be the corresponding clique hypergraph. In the terminology of Foulis and Randall, \(H\) is a manual of operations. The authors give a natural definition for a family of positive functions \(\mu _F\colon F\to {\mathbb{R}}^+\), \(F\in {\mathcal E}(G)\), to be compatible. Their main result states that \(G\) is triangulated if and only if any compatible family of positive functions \(\mu _F\), \(F\in {\mathcal E}(G)\), on \(H\) can be expressed in terms of a single (unnormed) measure on \(H\). The authors briefly discuss some applications to experimental data and random errors.
05C99 Graph theory
05C65 Hypergraphs
Full Text: EuDML
[1] BERAN L.: Orthomodular Lattices Algebraic Approach. Academia Press in co-edition with D. Reidel Publishing Company, Dordrecht, 1984. · Zbl 0558.06008
[2] BERGE C.: Graphs and Hypergraphs. North-Holland, Amsterdam, 1973. · Zbl 0254.05101
[3] CRAMÉR H.: Mathematical Methods of Statistics. Princeton, N.J., Princeton University Press, 1946. · Zbl 0063.01014
[4] DAVID A. P.-LAURITZEN S. L.: Hyper Markov laws in the statistical analysis of decomposable graphical models. Ann. Statist. 21 (1993), 1272-1317. · Zbl 0815.62038
[5] FINCH P. D.: Quantum mechanical physical quantities as random variables. Foundations of Probability Theory Statistical Inference, and Statistical Theories of Science, Vol. III (W. L. Harper, C A. Hooker, D. Reidel Publishing Company, Dordrecht-Holland, 1976, pp. 81-103. · Zbl 0349.02025
[6] FISCHER H. R.-RÜTTIMANN G. T.: The geometry of the state space. Mat\?ematical Found. of Quantum Theoru (A. Marlow, Academic Press, New York, 1978, pp. 153-177.
[7] FOULIS D. J.-RANDALL C. H.: Operational statistics. I. Basic concepts. J. Math. Phys. 13 (1972), 1667-1675. · Zbl 0287.60002
[8] GOLUMBIC M. C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, London, 1980. · Zbl 0541.05054
[9] GREECHIE R. J.: Orthomodular lattices admitting no states. J. Cornbin. Theory 10 (1971), 119-132. · Zbl 0219.06007
[10] GREECHIE R. J.-GUDDER S. P.: Quantum logics. Contemporary Research in the Foundations and Philosophy of Quantum Theory. (C. A. Hooker, Proc. Conf. Univ. Western Ontario, London, Canada, D. Reidel Pubhshing Companu, Dordrecht-Holland, 1973, pp. 143-173. · Zbl 0279.02015
[11] GUDDER S. P.-KLÄY M. P.-RÜTTIMANN G. T.: States on hypergraphs. Demonstratio Math. 19 (1986), 503-526.
[12] GUDDER S. P.-RÜTTIMANN G. T.: Observables an hypergraphs. Found. Phus. 16 (1986), 773-790.
[13] KLÄY M. P..: Quantum logic properties of hypergraphs. Found. Phys. 17 (1987), 1019-1036.
[14] LAURITZEN S. L.-SPEED T. P.-VIJAYAN K.: Decomposable graphs and hypergraphs. J. Austral. Math. Soc. Ser. A 36 (1984), 12-29. · Zbl 0533.05046
[15] LEIMER H.-G.: Triangulated graphs with marked vertices. Graph Theory in Memory of G. A. Dirac. (L. D. Andersen, C. Thomassen, B. Toft, P. D. Vestergaard, Ann. Discrete Math. 41, Elsevier Science Publishers B.V. (North-Holland), Amsterdam, 1989, pp. 311-324. · Zbl 0678.05056
[16] PASZKIEWICZ A.-SZYMANSKI A.: On some orthologics and posets determined by various orthogonality relations. 1994
[17] RANDALL C. H.-FOULIS D. J.: An approach to empirical logic. Amer. Math. Monthly 77 (1970), 363-374. · Zbl 0209.30302
[18] RÜTTIMANN G. T.: Jordan-Hahn decomposition of signed weights on finite orthogonality spaces. Comment. Math. Helv. 52 (1977), 129-144. · Zbl 0368.06008
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