Abstract
The topic of this survey are structures of stochastic conditional independence. Two basic questions are dealt with: the problem of characterization of conditional independence models and the problem of their mathematical description and computer representation. Basic formal properties of conditional independence are recapitulated and the problem of axiomatic characterization of stochastic conditional independence models is mentioned. Classic graphical methods of description of these structures are recalled, in particular, the method which uses chain graphs. Limitation of graphical approaches motivated an attempt at a non-graphical approach. A certain method of description of stochastic conditional independence models which uses non-graphical tools called ‘structural imsets’ is outlined.
Similar content being viewed by others
References
S.A. Andersson, D. Madigan and M.D. Perlman, A characterization ofMarkov equivalence for acyclic digraphs, Annals of Statistics 25 (1997) 505-541.
S.A. Andersson, D. Madigan and M.D. Perlman, Alternative Markov properties for chain graphs, Scandinavian Journal of Statistics 28 (2001) 33-85.
S.A. Andersson, D. Madigan, M.D. Perlman and T.S. Richardson, Graphical Markov models in multivariate analysis, in: Multivariate Analysis, Design of Experiments, and Survey Sampling, Statistical Textbooks Monographs, Vol. 159 (Dekker, 1999) pp. 187-229.
S. Benferhat, D. Dubois and H. Prade, Possibilistic independence and plausible reasoning, in: Advances in Fuzzy Systems-Applications and Theory, Vol. 8 (World Scientific, 1995) pp. 47-63.
B. Ben Yaghlane, P. Smets and K. Mellouli, Independence concepts for belief functions, in: Proceedings of 8th International Conference on Information Processing and Management of Uncertainty in Knowledge Based Systems, IPMU, Vol. I (2000) pp. 357-364.
R.R. Bouckaert and M. Studený, Chain graphs: semantics and expressiveness, in: Symbolic and Quantitative Approaches to Reasoning and Uncertainty, eds. Ch. Froidevaux and J. Kohlas, Lecture Notes in Artificial Intelligence, Vol. 946 (Springer, 1995) pp. 67-76.
G. Coletti and R. Scozzafava, Null events and stochastic independence, Kybernetika 34 (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, 2000) pp. 185-196.
I. Couso, S. Moral and P. Walley, A survey of concepts of independence for imprecise probabilities, Risk, Decision and Policy 5 (2000) 1-17.
R.G. Cowell, A.P. Dawid, S.L. Lauritzen and D.J. Spiegelhalter, Probabilistic Networks and Expert Systems (Springer, 1999).
D.R. Cox and N.Wermuth, Multivariate Dependencies-Models, Analysis, and Interpretation (Chapman and Hall, 1996).
A.P. Dawid, Conditional independence in statistical theory, Journal of Royal Statistical Society Series B 41 (1979) 1-31.
A.P. Dawid, Separoids: a general framework for conditional independence and irrelevance, to appear in Annals of Mathematics and Artificial Intelligence.
L.M. de Campos and J.F. Huete, Independence concepts in possibility theory, Parts I and II, Fuzzy Sets and Systems 103 (1999) 127-152, 487-505.
J.-P. Florens, M. Mouchart and J.-M. Rolin, Elements of Bayesian Statistics (Marcel Dekker, 1990).
M. Frydenberg, Marginalization and collapsability in graphical interaction models, Annals of Statistics 18 (1990) 790-805.
M. Frydenberg, The chain graph Markov property, Scandinavian Journal of Statistics 17 (1990) 333-353.
D. Geiger, T. Verma and J. Pearl, Identifying independence in Bayesian networks, Networks 20 (1990) 507-534.
D. Geiger and J. Pearl, On the logic of causal models, in: Uncertainty in Artificial Intelligence 4, eds. R.D. Shachter, T.S. Lewitt, L.N. Kanal and J.F. Lemmer (North-Holland, 1990) pp. 3-14.
D. Geiger and J. Pearl, Logical and algorithmic properties of conditional independence and their application to Bayesian networks, Annals of Mathematics and Artificial Intelligence 2 (1990) 165-178.
D. Geiger, A. Paz and J. Pearl, Axioms and algorithms for inferences involving probabilistic independence, Information and Computation 91 (1991) 128-141.
D. Geiger and J. Pearl, Logical and algorithmic properties of conditional independence and graphical models, Annals of Statistics 21 (1993) 2001-2021.
H. Kiiveri, T.P. Speed and J.B. Carlin, Recursive causal models, Journal of Australian Mathematical Society Series A 36 (1984) 30-52.
J.T.A. Koster, Markov properties of nonrecursive causal models, Annals of Statistics 24 (1996) 2148-2177.
S.L. Lauritzen and N. Wermuth, Mixed interaction models, Research report R-84-8, Institute of Electrical Systems, University of Aalborg (1984) (the research report was later modified and became a basis of the journal paper [26]).
S.L. Lauritzen and N. Wermuth, Graphical models for associations between variables, some of which are qualitative and some quantitative, Annals of Statistics 17 (1989) 31-57.
S.L. Lauritzen, Mixed graphical association models, Scandinavian Journal of Statistics 16 (1989) 273-306.
S.L. Lauritzen, A.P. Dawid, B.N. Larsen and H.-G. Leimer, Independence properties of directed Markov fields, Networks 20 (1990) 491-505.
S.L. Lauritzen, Graphical Models, Oxford Statistical Science Series, Vol. 17 (Clarendon Press, 1996).
F.M. Malvestuto, A unique formal system for binary decomposition of database relations, probability distributions and graphs, Information Sciences 59 (1992) 21-52; also F.M. Malvestuto and M. Studený, Comment on "A unique formal... graphs", Information Sciences 63 (1992) 1-2.
F.M. Malvestuto, A hypergraph-theoretic analysis of collapsibility and decomposability for extended log-linear models, Statistics and Computing 11 (2001) 155-169.
F. MatÚš, Stochastic independence, algebraic independence and abstract connectedness, Theoretical Computer Science 134 (1994) 455-471.
F. MatÚš and M. Studený, Conditional independencies among four random variables I, Combinatorics, Probability and Computing 4 (1995) 269-278.
F. MatÚš, Conditional independencies among four random variables II, Combinatorics, Probability and Computing 4 (1995) 407-417.
F. MatÚš, Conditional independencies among four random variables III. Final conclusion, Combinatorics, Probability and Computing 8 (1999) 269-276.
F. MatÚš, On the length of the semigraphoid inference, in: Proceedings of WUPES'2000, 5th Workshop on Uncertainty Processing, Jindřichův Hradec, Czech Republic (2000) pp. 176-180.
E. Mendelson, Introduction to Mathematical Logic (van Nostrand, 1979).
M. Mouchart and J.-M. Rolin, A note on conditional independence with statistical application, Statistica 44 (1984) 557-584.
J. Moussouris, Gibbs and Markov properties over undirected graphs, Journal of Statistical Physics 10 (1974) 11-31.
K.K. Nambiar, Some analytic tools for the design of relational database systems, in: Proceedings of the 6th Conference on Very Large Data Bases (1980) pp. 417-428.
A. Paz, R.Y. Geva and M. Studený, Representation of irrelevance relations by annotated graphs, Fundamenta Informaticae 42 (2000) 149-199.
J. Pearl and A. Paz, Graphoids: a graph-based logic for reasoning about relevance relations, in: Advances in Artificial Intelligence II, eds. B. Du Boulay, D. Hogg and L. Steels (North-Holland, 1987) pp. 357-363.
J. Pearl, Probabilistic Reasoning in Intelligent Systems, Networks of Plausible Inference (Morgan Kaufmann, 1988).
W. Rudin, Real and Complex Analysis (McGraw-Hill, 1974).
R. Scozzafava and B. Vantaggi (eds.), Extended Abstracts of the Conference "Partial Knowledge and Uncertainty: Independence, Conditioning, Inference", Rome, Italy (2000).
P.P. Shenoy, Conditional independence in valuation-based systems, International Journal of Approximate Reasoning 10 (1994) 203-234.
J.Q. Smith, Influence diagrams for statistical modelling, Annals of Statistics 17 (1989) 654-672.
P. Spirtes, Directed cyclic graphical representations of feedback models, in: Uncertainty in Artificial Intelligence 11, eds. P. Besnard and S. Hanks (Morgan Kaufmann, 1995) pp. 491-498.
W. Spohn, Stochastic independence, causal independence and shieldability, Journal of Philosophical Logic 9 (1980) 73-99.
M. Studený, Multiinformation and the problem of characterization of conditional independence relations, Problems of Control and Information Theory 18 (1989) 3-16.
M. Studený, Conditional independence relations have no finite complete characterization, in: Information Theory, Statistical Decision Functions and Random Processes, Vol. B, eds. S. Kubík and J.Á. Víšek (Kluwer/Academia, 1992) pp. 377-396.
M. Studený, Formal properties of conditional independence in different calculi of AI, in: Symbolic and Quantitative Approaches to Reasoning and Uncertainty, eds. M. Clarke, R. Kruse and S. Moral, Lecture Notes in Computer Science, Vol. 747 (Springer, 1993) pp. 341-348.
M. Studený, Description of conditional independence structures by means of imsets: a connection with product formula validity, in: Uncertainty in Intelligent Systems, eds. B. Bouchon-Meunier, L.L. Valverde and R.R. Yager (Elsevier, 1993) pp. 179-194.
M. Studený, Description of structures of stochastic conditional independence by means of faces and imsets, 1st Part: introduction and basic concepts; 2nd Part: basic theory; 3rd Part: examples of use and appendices, International Journal of General Systems 23 (1994/1995) 123-137, 201-219, 323-341.
M. Studený, Semigraphoids are two-antecedental approximations of stochastic conditional independence models, in: Uncertainty in Artificial Intelligence 10, eds. R.L. de Mantaras and D. Poole (Morgan Kaufmann, 1994) pp. 546-552.
M. Studený and P. Boček, CI-models arising among 4 random variables, in: Proceedings of WUPES' 94 (3rd Workshop on Uncertainty Processing in Expert Systems), Třešt, Czech Republic (1994) pp. 268-282.
M. Studený, Semigraphoids and structures of probabilistic conditional independence, Annals of Mathematics and Artificial Intelligence 21 (1997) 71-98.
M. Studený, A recovery algorithm for chain graphs, International Journal of Approximate Reasoning 17 (1997) 265-293.
M. Studený and R.R. Bouckaert, On chain graph models for description of conditional independence structures, Annals of Statistics 26 (1998) 1434-1495.
M. Studený, On mathematical description of probabilistic conditional independence structures, Thesis for DrSc degree, Institute of Information Theory and Automation, Prague (May 2001) (a survey monograph).
J. Vejnarová, Conditional independence relations in possibility theory, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems 8 (2000) 253-269.
T. Verma and J. Pearl, Causal netwoks: semantics and expressiveness, in: Uncertainty in Artificial Intelligence 4, eds. R.D. Shachter, T.S. Lewitt, L.N. Kanal and J.F. Lemmer (North-Holland, 1990) pp. 69-76.
Y. Xiang, S.K.M.Wong and N. Cercone, Critical remarks on single link search in learning belief networks, in: Uncertainty in Artificial Intelligence 12, eds. E. Horvitz and F. Jensen (Morgan Kaufmann, 1996) pp. 564-571.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Studený, M. On Stochastic Conditional Independence: the Problems of Characterization and Description. Annals of Mathematics and Artificial Intelligence 35, 323–341 (2002). https://doi.org/10.1023/A:1014559906385
Issue Date:
DOI: https://doi.org/10.1023/A:1014559906385