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2017 | OriginalPaper | Buchkapitel

On Conditioning in Multidimensional Probabilistic Models

verfasst von : Radim Jiroušek

Erschienen in: Robustness in Econometrics

Verlag: Springer International Publishing

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Abstract

Graphical Markov models, and above all Bayesian networks have become a very popular tool for multidimensional probability distribution representation and processing. The technique making computation with several hundred dimensional probability distribution possible was suggested by Lauritzen and Spiegelhalter. However, to employ it one has to transform a Bayesian network into a decomposable model. This is because decomposable models (or more precisely their building blocks, i.e., their low-dimensional marginals) can be reordered in many ways, so that each variable can be placed at the beginning of the model. It is not difficult to show that there is a much wider class of models possessing this property. In compositional models theory we call these models flexible. It is the widest class of models for which one can always restructure the model in the way that any variable can appear at the beginning of the model. But until recently it had been an open problem whether this class of models is closed under conditioning; i.e., whether a conditional of a flexible model is again flexible. In the paper we will show that this property holds true, which proves the importance of flexible models for practical applications.

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Fußnoten
1
The reader interested in marginalization procedures is referred to algorithms designed for computations with compositional models either in Malvestuto’s papers [15, 16] and/or in [2].
 
Literatur
1.
2.
Zurück zum Zitat Bína V, Jiroušek R (2006) Marginalization in multidimensional compositional models. Kybernetika 42(4):405–422MathSciNetMATH Bína V, Jiroušek R (2006) Marginalization in multidimensional compositional models. Kybernetika 42(4):405–422MathSciNetMATH
3.
Zurück zum Zitat Bína V, Jiroušek R (2015) On computations with causal compositional models. Kybernetika 51(3):525–539MathSciNetMATH Bína V, Jiroušek R (2015) On computations with causal compositional models. Kybernetika 51(3):525–539MathSciNetMATH
4.
Zurück zum Zitat Dawid AP (1992) Applications of a general propagation algorithm for probabilistic expert systems. Stat Comput 2(1):25–26CrossRef Dawid AP (1992) Applications of a general propagation algorithm for probabilistic expert systems. Stat Comput 2(1):25–26CrossRef
5.
Zurück zum Zitat Jensen FV (2001) Bayesian networks and decision graphs. IEEE Computer Society Press, New York Jensen FV (2001) Bayesian networks and decision graphs. IEEE Computer Society Press, New York
6.
Zurück zum Zitat Jiroušek R (2002) Decomposition of multidimensional distributions represented by perfect sequences. Ann Math Artif Intell 35(1–4):215–226MathSciNetCrossRefMATH Jiroušek R (2002) Decomposition of multidimensional distributions represented by perfect sequences. Ann Math Artif Intell 35(1–4):215–226MathSciNetCrossRefMATH
9.
Zurück zum Zitat Jiroušek R, (2013) Brief introduction to probabilistic compositional models. In: Huynh VN, Kreinovich V, Sriboonchita S, Suriya K (eds) Uncertainty analysis in econometrics with applications AISC 200. Springer, Berlin, pp 49–60 Jiroušek R, (2013) Brief introduction to probabilistic compositional models. In: Huynh VN, Kreinovich V, Sriboonchita S, Suriya K (eds) Uncertainty analysis in econometrics with applications AISC 200. Springer, Berlin, pp 49–60
10.
Zurück zum Zitat Jiroušek R, (2016) Brief introduction to causal compositional models. In: Huynh VN, Kreinovich V, Sriboonchita S (eds) Causal inference in econometrics. Studies in computational intelligence (SCI), vol 622. Springer, Cham, pp 199–211 Jiroušek R, (2016) Brief introduction to causal compositional models. In: Huynh VN, Kreinovich V, Sriboonchita S (eds) Causal inference in econometrics. Studies in computational intelligence (SCI), vol 622. Springer, Cham, pp 199–211
11.
12.
Zurück zum Zitat Lauritzen SL (1996) Graphical models. Oxford University Press, OxfordMATH Lauritzen SL (1996) Graphical models. Oxford University Press, OxfordMATH
13.
Zurück zum Zitat Lauritzen SL, Spiegelhalter D (1988) Local computation with probabilities on graphical structures and their application to expert systems. J R Stat Soc B 50:157–224MathSciNetMATH Lauritzen SL, Spiegelhalter D (1988) Local computation with probabilities on graphical structures and their application to expert systems. J R Stat Soc B 50:157–224MathSciNetMATH
14.
Zurück zum Zitat Madsen AL, Jensen F, Kjærulff UB, Lang M (2005) HUGIN-the tool for bayesian networks and influence diagrams. Int J Artif Intell Tools 14(3):507–543CrossRefMATH Madsen AL, Jensen F, Kjærulff UB, Lang M (2005) HUGIN-the tool for bayesian networks and influence diagrams. Int J Artif Intell Tools 14(3):507–543CrossRefMATH
15.
Zurück zum Zitat Malvestuto FM (2014) Equivalence of compositional expressions and independence relations in compositional models. Kybernetika 50(3):322–362MathSciNetMATH Malvestuto FM (2014) Equivalence of compositional expressions and independence relations in compositional models. Kybernetika 50(3):322–362MathSciNetMATH
16.
Zurück zum Zitat Malvestuto FM (2015) Marginalization in models generated by compositional expressions. Kybernetika 51(4):541–570MathSciNetMATH Malvestuto FM (2015) Marginalization in models generated by compositional expressions. Kybernetika 51(4):541–570MathSciNetMATH
17.
Zurück zum Zitat Pearl J (1982) Reverend Bayes on inference engines: a distributed hierarchical approach. In: Proceedings of the national conference on artificial intelligence, Pittsburgh, pp 133–136 Pearl J (1982) Reverend Bayes on inference engines: a distributed hierarchical approach. In: Proceedings of the national conference on artificial intelligence, Pittsburgh, pp 133–136
18.
Zurück zum Zitat Pearl J (2014) Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann, Burlington (Revised Second Printing)MATH Pearl J (2014) Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann, Burlington (Revised Second Printing)MATH
20.
Zurück zum Zitat Shachter RD (1988) Probabilistic inference and influence diagrams. Oper Res 36(4):589–604CrossRefMATH Shachter RD (1988) Probabilistic inference and influence diagrams. Oper Res 36(4):589–604CrossRefMATH
21.
Zurück zum Zitat Shenoy PP, Shafer G (2008) Axioms for probability and belief-function propagation. In: Classic works of the dempster-shafer theory of belief functions. Springer, Berlin, pp 499–528 Shenoy PP, Shafer G (2008) Axioms for probability and belief-function propagation. In: Classic works of the dempster-shafer theory of belief functions. Springer, Berlin, pp 499–528
Metadaten
Titel
On Conditioning in Multidimensional Probabilistic Models
verfasst von
Radim Jiroušek
Copyright-Jahr
2017
DOI
https://doi.org/10.1007/978-3-319-50742-2_12