What Cannot be Learned with Bethe Approximations

Uri Heinemann, Amir Globerson
Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence, PMLR R9:357-364, 2011.

Abstract

We address the problem of learning the parameters in graphical models when inference is intractable. A common strategy in this case is to replace the partition function with its Bethe approximation. We show that there exists a regime of empirical marginals where such Bethe learning will fail. By failure we mean that the empirical marginals cannot be recovered from the approximated maximum likelihood parameters (i.e., moment matching is not achieved). We provide several conditions on empirical marginals that yield outer and inner bounds on the set of Bethe learnable marginals. An interesting implication of our results is that there exists a large class of marginals that cannot be obtained as stable fixed points of belief propagation. Taken together our results provide a novel approach to analyzing learning with Bethe approximations and highlight when it can be expected to work or fail.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR9-heinemann11a, title = {What Cannot be Learned with {B}ethe Approximations}, author = {Heinemann, Uri and Globerson, Amir}, booktitle = {Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence}, pages = {357--364}, year = {2011}, editor = {Cozman, Fabio and Pfeffer, Avi}, volume = {R9}, series = {Proceedings of Machine Learning Research}, month = {14--17 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r9/main/assets/heinemann11a/heinemann11a.pdf}, url = {https://proceedings.mlr.press/r9/heinemann11a.html}, abstract = {We address the problem of learning the parameters in graphical models when inference is intractable. A common strategy in this case is to replace the partition function with its Bethe approximation. We show that there exists a regime of empirical marginals where such Bethe learning will fail. By failure we mean that the empirical marginals cannot be recovered from the approximated maximum likelihood parameters (i.e., moment matching is not achieved). We provide several conditions on empirical marginals that yield outer and inner bounds on the set of Bethe learnable marginals. An interesting implication of our results is that there exists a large class of marginals that cannot be obtained as stable fixed points of belief propagation. Taken together our results provide a novel approach to analyzing learning with Bethe approximations and highlight when it can be expected to work or fail.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T What Cannot be Learned with Bethe Approximations %A Uri Heinemann %A Amir Globerson %B Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2011 %E Fabio Cozman %E Avi Pfeffer %F pmlr-vR9-heinemann11a %I PMLR %P 357--364 %U https://proceedings.mlr.press/r9/heinemann11a.html %V R9 %X We address the problem of learning the parameters in graphical models when inference is intractable. A common strategy in this case is to replace the partition function with its Bethe approximation. We show that there exists a regime of empirical marginals where such Bethe learning will fail. By failure we mean that the empirical marginals cannot be recovered from the approximated maximum likelihood parameters (i.e., moment matching is not achieved). We provide several conditions on empirical marginals that yield outer and inner bounds on the set of Bethe learnable marginals. An interesting implication of our results is that there exists a large class of marginals that cannot be obtained as stable fixed points of belief propagation. Taken together our results provide a novel approach to analyzing learning with Bethe approximations and highlight when it can be expected to work or fail. %Z Reissued by PMLR on 04 October 2026.
APA
Heinemann, U. & Globerson, A.. (2011). What Cannot be Learned with Bethe Approximations. Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R9:357-364 Available from https://proceedings.mlr.press/r9/heinemann11a.html. Reissued by PMLR on 04 October 2026.

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