×

A universal algebraic approach for conditional independence. (English) Zbl 1440.62030

Summary: In this paper we show that elementary properties of joint probability density functions naturally induce a universal algebraic structure suitable for studying probabilistic conditional independence (PCI) relations. We call this algebraic system the cain. In the cain algebra, PCI relations are represented in equational forms. In particular, we show that the cain satisfies the axioms of the graphoid of J. Pearl and A. Paz [“Graphoids: a graph-based logic for reasoning about relevance relations”, in: D. Hogg (ed.) and L. Steels (ed.), Advances in artificial intelligence. Amsterdam: North-Holland. 357–363 (1987)] and the separoid of A. P. Dawid [Ann. Math. Artif. Intell. 32, No. 1–4, 335–372 (2001; Zbl 1314.68308)], these axiomatic systems being useful for general probabilistic reasoning.

MSC:

62A01 Foundations and philosophical topics in statistics
08A70 Applications of universal algebra in computer science
62H99 Multivariate analysis

Citations:

Zbl 1314.68308

Software:

Separoids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andersson S.A., Madsen J. (1998) Symmetry and lattice conditional independence in a multivariate normal distribution. Annals of Statistics, 26: 525–572 · Zbl 0943.62047 · doi:10.1214/aos/1028144848
[2] Andersson S.A., Perlman M.D. (1993) Lattice models for conditional independence in a multivariate normal distribution. Annals of Statistics, 21: 1318–1358 · Zbl 0803.62042 · doi:10.1214/aos/1176349261
[3] Andersson S.A., Perlman M.D. (1995) Testing lattice conditional independence models. Journal of Multivariate Analysis, 53: 18–38 · Zbl 0877.62057 · doi:10.1006/jmva.1995.1022
[4] Andersson S.A., Perlman M.D. (1995) Unbiasedness for the likelihood ratio test for lattice conditional independence models. Journal of Multivariate Analysis, 53: 1–17 · Zbl 0877.62058 · doi:10.1006/jmva.1995.1021
[5] Andersson S.A., Madigan D., Perlman M.D., Triggs C.M. (1995) On the relation between conditional independence models determined by finite distributive lattices and by directed acyclic graphs. Journal of Statistical Planning and Inference, 48: 25–46 · Zbl 0839.62063 · doi:10.1016/0378-3758(94)00150-T
[6] Andersson S.A., Madigan D., Perlman M.D., Triggs C.M. (1997) A graphical characterization of lattice conditional independence models. Annals of Mathematics and Artificial Intelligence 21: 27–50 · Zbl 0888.68090 · doi:10.1023/A:1018901032102
[7] Birkhoff, G. (1967). Lattice theory (Vol. 25, 3rd ed.). New York: American Mathematical Society Colloquium Publications, American Mathematical Society. · Zbl 0153.02501
[8] Blyth T.S. (2005) Lattices and ordered algebraic structures. Springer, London · Zbl 1073.06001
[9] Cox D., Little J., O’Shea D. (1997) Ideals, varieties and algorithms (2nd ed). Springer, New York
[10] Davey B.A., Priestley H.A. (2002) Introduction to lattices and order. Cambridge University Press, Cambridge · Zbl 1002.06001
[11] Dawid A.P. (1979) Conditional independence in statistical theory (with discussion). Journal of the Royal Statistical Society: Series B 41: 1–31 · Zbl 0408.62004
[12] Dawid A.P. (1979) Some misleading arguments involving conditional independence. Journal of the Royal Statistical Society: Series B, 41: 249–252
[13] Dawid A.P. (1980) A Bayesian look at nuisance parameters. In: Bernardo J.M., DeGroot M.H., Lindley D.V., Smith A.F.M. (Eds) (eds) Bayesian statistics.. Valencia University Press, Valencia, pp 167–184 · Zbl 0455.62004
[14] Dawid A.P. (1980) Conditional independence for statistical operations. Annals of Statistics, 8: 598–617 · Zbl 0434.62006 · doi:10.1214/aos/1176345011
[15] Dawid A.P. (1985) Invariance and independence in multivariate distribution theory. Journal of Multivariate Analysis, 17: 304–315 · Zbl 0602.62038 · doi:10.1016/0047-259X(85)90086-7
[16] Dawid A.P. (1988) Conditional independence. In: Kotz S., Read C.B., Banks D.L.(Eds) (eds) Encyclopedia of statistical science (Vol. 2). Wiley, New York, pp 146–155
[17] Dawid A.P. (2001) Separoids: A mathematical framework for conditional independence and irrelevance. Annals of Mathematics and Artificial Intelligence, 32: 335–372 · Zbl 1314.68308 · doi:10.1023/A:1016734104787
[18] Dawid A.P., Evett I.W. (1997) Using a graphical method to assist the evaluation of complicated patterns of evidence. Journal of Forensic Sciences, 42: 226–231
[19] Dawid A.P., Lauritzen S.L. (1993) Hyper Markov laws in the statistical analysis of decomposable graphical models. Annals of Statistics, 21: 1272–1317 · Zbl 0815.62038 · doi:10.1214/aos/1176349260
[20] Dawid A.P., Mortera J. (1996) Coherent analysis of forensic identification evidence. Journal of the Royal Statistical Society: Series B, 58: 425–443 · Zbl 0853.62095
[21] Dawid A.P., Studený M. (1999) Conditional products: an alternative approach to conditional independence. In: Heckerman D., Whittaker J.(Eds) (eds) Artificial intelligence and statistics (Vol 99).. Morgan Kaufmann, San Francisco, pp 32–40
[22] Freese R., Jezek J., Nation J.B. (1995) Free lattices. Mathematical surveys and monographs. American Mathematical Society, Providence
[23] Galles D., Pearl J. (1997) Axioms of causal relevance. Artificial Intelligence, 97: 9–43 · Zbl 0917.68123 · doi:10.1016/S0004-3702(97)00047-7
[24] Halmos P.R. (1974) Lectures on boolean algebras. Springer, New York · Zbl 0285.06010
[25] Jipsen, P., Rose, H. (1992). Varieties of lattices. Lecture notes in mathematics (Vol. 1533). Springer: New York. http://www1.chapman.edu/\(\sim\)jipsen/JipsenRoseVoL.html . · Zbl 0779.06005
[26] Jónsson B., Kiefer J. (1962) Finite sublattices of a free lattice. Canadian Journal of Mathematics, 14: 487–497 · Zbl 0107.25202 · doi:10.4153/CJM-1962-040-1
[27] Lauritzen S.L. (1982) Lectures on contingency tables (2nd ed). University of Aalborg Press, Aalborg
[28] Lauritzen S.L. (1996) Graphical models. Clarendon, Oxford · Zbl 0907.62001
[29] Lauritzen S.L., Dawid A.P., Larsen B.N., Leimer H.G. (1990) Independence properties of directed Markov fields. Networks, 20: 491–505 · Zbl 0743.05065 · doi:10.1002/net.3230200503
[30] Massam H., Neher E. (1998) Estimation and testing for lattice conditional independence models on Euclidean Jordan algebras. Annals of Statistics, 26: 1051–1082 · Zbl 0932.62067 · doi:10.1214/aos/1024691088
[31] Nation, J. B. Notes on lattice, unpublished course notes. http://www.math.hawaii.edu/\(\sim\)jb/lat1-6.pdf .
[32] Paz, A., Pearl, J. (1994). Axiomatic characterization of directed graphs. Technical Report R-234. Los Angeles: Cognitive Systems Laboratory, Computer Science Department, University of California.
[33] Paz A., Pearl J., Ur S. (1996) A new characterization of graphs based on interception relations. Journal of Graph Theory, 22: 125–136 · Zbl 0859.05066 · doi:10.1002/(SICI)1097-0118(199606)22:2<125::AID-JGT3>3.0.CO;2-P
[34] Pearl J. (1988) Probabilistic reasoning in intelligent systems. Morgan Kaufmann, San Mateo · Zbl 0746.68089
[35] Pearl J. (2000) Causality–models, reasoning and inference. Cambridge University Press, New York · Zbl 0959.68116
[36] Pearl J., Paz A. (1987) Graphoids: a graph-based logic for reasoning about relevance relations. In: Hogg D., Steels L.(Eds) (eds) Advances in artificial intelligence.. North-Holland, Amsterdam, pp 357–363
[37] Reichenbach H. (1956) The direction of time. University of California Press, Berkeley
[38] Sagiv Y., Walecka S.F. (1982) Subset dependencies and completeness result for a subclass of embedded multivalued dependencies. Journal of the Association for Computing Machinery, 29: 103–117 · Zbl 0485.68091 · doi:10.1145/322290.322297
[39] Salii V.N. (1988) Lattices with unique complements. Translations of mathematical monographs. American Mathematical Society, Providence
[40] Schoken S.S., Hummel R.A. (1993) On the use of the dempster shafer model in information indexing and retrieval applications. International Journal of Man–Machine Studies, 39: 1–37 · doi:10.1006/imms.1993.1051
[41] Shafer G. (1976) A Mathematical theory of evidence. Princeton University Press, Princeton · Zbl 0359.62002
[42] Shenoy P.P. (1994) Conditional independence in valuation-based systems. International Journal of Approximate Reasoning, 10: 203–234 · Zbl 0821.68114 · doi:10.1016/0888-613X(94)90001-9
[43] Spohn W. (1980) Stochastic independence, causal independence, and shieldability. Journal of Philosophical Logic 9: 73–99 · Zbl 0436.60004 · doi:10.1007/BF00258078
[44] Spohn W. (1988) Ordinal conditional functions: a dynamic theory of epistemic states. In: Harper W.L., Skyrms B.(Eds) (eds) Causation in decision, belief change, and statistics (Vol. 2). Kluwer, Dordrecht, pp 105–134
[45] Spohn, W. (1994). On the properties of conditional independence. In P. Humphreys (Ed.), Patrick Suppes: scientific philosopher. Probability and probabilistic causality (Vol. 1, pp. 173–194). Dordrecht: Kluwer.
[46] Stanley, B. N., Sankappanavar, H. P. (1981). A course in universal algebra. New York: Springer. http://www.math.uwaterloo.ca/\(\sim\)snburris/htdocs/ualg.html . · Zbl 0478.08001
[47] Studený, M. (1993). Formal properties of conditional independence in different calculi of AI. In Symbolic and quantitative approaches to reasoning and uncertainty. Lecture notes in computer science (Vol. 747, pp. 341–348). Berlin: Springer.
[48] Studený M. (2005) Probabilistic conditional independence structures (Vol 97). Springer, London · Zbl 1070.62001
[49] Teixeirade Silva W., Milidiu R.L. (1993) Belief function model for information retrieval. Journal of the American Society for Information Science, 44: 10–18 · doi:10.1002/(SICI)1097-4571(199301)44:1<10::AID-ASI2>3.0.CO;2-V
[50] Zadeh L.A. (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1: 3–28 · Zbl 0377.04002 · doi:10.1016/0165-0114(78)90029-5
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.