| Title:
|
Compositional models, Bayesian models and recursive factorization models (English) |
| Author:
|
Malvestuto, Francesco M. |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
52 |
| Issue:
|
5 |
| Year:
|
2016 |
| Pages:
|
696-723 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Compositional models are used to construct probability distributions from lower-order probability distributions. On the other hand, Bayesian models are used to represent probability distributions that factorize according to acyclic digraphs. We introduce a class of models, called \emph{recursive factorization models}, to represent probability distributions that recursively factorize according to sequences of sets of variables, and prove that they have the same representation power as both compositional models generated by sequential expressions and Bayesian models. Moreover, we present a linear (graphical) algorithm for deciding if a conditional independence is valid in a given recursive factorization model. (English) |
| Keyword:
|
Bayesian model |
| Keyword:
|
compositional model |
| Keyword:
|
conditional independence |
| Keyword:
|
Markov property |
| Keyword:
|
recursive model |
| Keyword:
|
sequential expression |
| MSC:
|
05C65 |
| MSC:
|
05C85 |
| MSC:
|
60E99 |
| MSC:
|
65C50 |
| MSC:
|
68T37 |
| idZBL:
|
Zbl 06674935 |
| idMR:
|
MR3602011 |
| DOI:
|
10.14736/kyb-2016-5-0696 |
| . |
| Date available:
|
2017-01-02T13:25:06Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145964 |
| . |
| Reference:
|
[1] Aho, A. V., Hopcroft, J. E., Ullman, J. D.: Data Structures and Algorithms..Addison-Wesley Pub. Co, Reading 1987. Zbl 0487.68005, MR 0666695 |
| Reference:
|
[2] Bína, V., Jiroušek, R.: On computations with causal compositional models..Kybernetika 51 (2015), 525-539. Zbl 1340.65011, MR 3391683, 10.14736/kyb-2015-3-0525 |
| Reference:
|
[3] Geiger, D., Verma, T., Pearl, J.: Identifying independence in Bayesian networks..Networks 20 (1990), 507-534. Zbl 0724.05066, MR 1064736, 10.1002/net.3230200504 |
| Reference:
|
[4] Jiroušek, R.: Composition of probability measures on finite spaces..In: Proc. XIII International Conference on Uncertainty in Artificial Intelligence (D. Geiger and P. P. Shenoy, eds.), Morgan Kaufmann, San Francisco 1997, pp. 274-281. 10.1023/a:1014591402750 |
| Reference:
|
[5] Jiroušek, R.: Decomposition of multidimensional distributions represented by perfect sequences..Ann. Math. Artif. Intelligence 5 (2002), 215-226. Zbl 1004.60010, MR 1899952, 10.1023/a:1014591402750 |
| Reference:
|
[6] Jiroušek, R.: Foundations of compositional model theory..Int. J. General Systems 40 (2011), 623-678. Zbl 1252.68285, MR 2817988, 10.1080/03081079.2011.562627 |
| Reference:
|
[7] Jiroušek, R.: Local computations in Dempster-Shafer theory of evidence..Int. J. Approx. Reasoning 53 (2012), 1155-1167. Zbl 1266.68177, MR 2971864, 10.1016/j.ijar.2012.06.012 |
| Reference:
|
[8] Jiroušek, R.: On causal compositional models: simple examples..In: Proc. XIV International Conference on Information Processing and Management of Uncertainty in Knowledge-Bases Systems (IPMU 2014) (A. Laurent et al., eds.), Part I, CCIS 442, pp. 517-526. 10.1007/978-3-319-08795-5_53 |
| Reference:
|
[9] Jiroušek, R., Kratochvíl, V.: Foundations of compositional models: structural properties..Int. J. General Systems 44 (2015), 2-25. MR 3299901, 10.1080/03081079.2014.934370 |
| Reference:
|
[10] Jiroušek, R., Shenoy, P. P.: Compositional models in valuation-based systems..Int. J. Approx. Reasoning 55 (2014), 277-293. Zbl 1252.68310, MR 3133554, 10.1016/j.ijar.2013.02.002 |
| Reference:
|
[11] Jiroušek, R., Vejnarová, J.: General framework for multidimensional models..Int. J. Approx. Reasoning 18 (2003), 107-127. Zbl 1029.68131, 10.1002/int.10077 |
| Reference:
|
[12] Jiroušek, R., Vejnarová, J., Daniels, M.: Composition models of belief functions..In: Proc. V Symp. on Imprecise Probabilities and Their Applications (G. De Cooman, J. Vejnarová and M. Zaffalon, eds.), Action M Agency, Prague 2007, pp. 243-252. |
| Reference:
|
[13] Kratochvíl, V.: Characteristic properties of equivalent structures in compositional models..Int. J. of Approx. Reasoning 52 (2011), 599-612. Zbl 1214.68400, MR 2787020, 10.1016/j.ijar.2010.12.005 |
| Reference:
|
[14] Kratochvíl, V.: Probabilistic compositional models: solutions of an equivalence problem..Int. J. Approx. Reasoning 54 (2013), 590-601. MR 3041095, 10.1016/j.ijar.2013.01.002 |
| Reference:
|
[15] Lauritzen, S. L.: Graphical Models..Oxford University Press, Oxford 1996. MR 1419991 |
| Reference:
|
[16] Lauritzen, S. L., Dawid, A. P., Larsen, B. N., Leimer, H.-G.: Independence properties of directed Markov fields..Networks 20 (1990), 491-505. Zbl 0743.05065, MR 1064735, 10.1002/net.3230200503 |
| Reference:
|
[17] Maier, D.: The Theory of Relational Databases..Computer Science Press, 1983. Zbl 0519.68082, MR 0691493 |
| Reference:
|
[18] Malvestuto, F. M.: A join-like operator to combine data cubes, and answer queries from multiple data cubes..ACM Trans. Database Systems 39, (2014), 3, 1-31. MR 3268995, 10.1145/2638545 |
| Reference:
|
[19] Malvestuto, F. M.: Equivalence of compositional expressions and independence relations in compositional models..Kybernetika 50 (2014), 322-362. MR 3245534, 10.14736/kyb-2014-3-0322 |
| Reference:
|
[20] Malvestuto, F. M.: Marginalization in models generated by compositional expressions..Kybernetika 51 (2015), 541-570. MR 3423187, 10.14736/kyb-2015-4-0541 |
| Reference:
|
[21] Malvestuto, F. M.: A general composition operator for probabilistic compositional models..Unpublished manuscript (2016). |
| Reference:
|
[22] Pearl, J.: Probabilistic Reasoning in Intelligent Systems..Morgan Kaufmann Pub., San Mateo 1988. Zbl 0746.68089, MR 0965765 |
| Reference:
|
[23] Pourabbas, E., Shoshani, A.: Efficient estimation of joint queries from multiple OLAP databases..ACM Trans. Database Systems 32 (2007), 1, Article No. 2. 10.1145/1206049.1206051 |
| Reference:
|
[24] Tarjan, R. E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs..SIAM J. Comput. 13 (1984), 566-579. Zbl 0562.68055, MR 0749707, 10.1137/0213035 |
| Reference:
|
[25] Verma, T., Pearl, J.: Causal networks: semantics and expressiveness..In: Proc. IV Workshop on Uncertainty in Artificial Intelligence, St. Paul 1988, pp. 352-359. |
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