Representations of Bayesian networks by low-rank models

Petr Tichavský, Jiří Vomlel
Proceedings of the Ninth International Conference on Probabilistic Graphical Models, PMLR 72:463-474, 2018.

Abstract

Conditional probability tables (CPTs) of discrete valued random variables may achieve high dimensions and Bayesian networks defined as the product of these CPTs may become intractable by conventional methods of BN inference because of their dimensionality. In many cases, however, these probability tables constitute tensors of relatively low rank. Such tensors can be written in the so-called Kruskal form as a sum of rank-one components. Such representation would be equivalent to adding one artificial parent to all random variables and deleting all edges between the variables. The most difficult task is to find such a representation given a set of marginals or CPTs of the random variables under consideration. In the former case, it is a problem of joint canonical polyadic (CP) decomposition of a set of tensors. The latter fitting problem can be solved in a similar manner. We apply a recently proposed alternating direction method of multipliers (ADMM), which assures that the model has a probabilistic interpretation, i.e., that all elements of all factor matrices are nonnegative. We perform experiments with several well-known Bayesian networks.

Cite this Paper

BibTeX
@InProceedings{pmlr-v72-tichavsky18a, title = {Representations of Bayesian networks by low-rank models}, author = {Tichavsk\'{y}, Petr and Vomlel, Ji\v{r}\'{i}}, booktitle = {Proceedings of the Ninth International Conference on Probabilistic Graphical Models}, pages = {463--474}, year = {2018}, editor = {Kratochvíl, Václav and Studený, Milan}, volume = {72}, series = {Proceedings of Machine Learning Research}, month = {11--14 Sep}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v72/tichavsky18a/tichavsky18a.pdf}, url = {https://proceedings.mlr.press/v72/tichavsky18a.html}, abstract = {Conditional probability tables (CPTs) of discrete valued random variables may achieve high dimensions and Bayesian networks defined as the product of these CPTs may become intractable by conventional methods of BN inference because of their dimensionality. In many cases, however, these probability tables constitute tensors of relatively low rank. Such tensors can be written in the so-called Kruskal form as a sum of rank-one components. Such representation would be equivalent to adding one artificial parent to all random variables and deleting all edges between the variables. The most difficult task is to find such a representation given a set of marginals or CPTs of the random variables under consideration. In the former case, it is a problem of joint canonical polyadic (CP) decomposition of a set of tensors. The latter fitting problem can be solved in a similar manner. We apply a recently proposed alternating direction method of multipliers (ADMM), which assures that the model has a probabilistic interpretation, i.e., that all elements of all factor matrices are nonnegative. We perform experiments with several well-known Bayesian networks.} }
Endnote
%0 Conference Paper %T Representations of Bayesian networks by low-rank models %A Petr Tichavský %A Jiří Vomlel %B Proceedings of the Ninth International Conference on Probabilistic Graphical Models %C Proceedings of Machine Learning Research %D 2018 %E Václav Kratochvíl %E Milan Studený %F pmlr-v72-tichavsky18a %I PMLR %P 463--474 %U https://proceedings.mlr.press/v72/tichavsky18a.html %V 72 %X Conditional probability tables (CPTs) of discrete valued random variables may achieve high dimensions and Bayesian networks defined as the product of these CPTs may become intractable by conventional methods of BN inference because of their dimensionality. In many cases, however, these probability tables constitute tensors of relatively low rank. Such tensors can be written in the so-called Kruskal form as a sum of rank-one components. Such representation would be equivalent to adding one artificial parent to all random variables and deleting all edges between the variables. The most difficult task is to find such a representation given a set of marginals or CPTs of the random variables under consideration. In the former case, it is a problem of joint canonical polyadic (CP) decomposition of a set of tensors. The latter fitting problem can be solved in a similar manner. We apply a recently proposed alternating direction method of multipliers (ADMM), which assures that the model has a probabilistic interpretation, i.e., that all elements of all factor matrices are nonnegative. We perform experiments with several well-known Bayesian networks.
APA
Tichavský, P. & Vomlel, J.. (2018). Representations of Bayesian networks by low-rank models. Proceedings of the Ninth International Conference on Probabilistic Graphical Models, in Proceedings of Machine Learning Research 72:463-474 Available from https://proceedings.mlr.press/v72/tichavsky18a.html.

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