Abstract
Aggregating a set of Bayesian Networks (BNs), also known as BN fusion, has been studied in the literature, providing a precise theoretical framework for the structural phase. This phase depends on a total ordering of the variables, but both the problem of searching for the optimal consensus structure (according to standard problem definition), as well as the one of looking for the optimal ordering are NP-hard.
In this paper we start from this theoretical framework and extend it from a practical point of view. We propose a heuristic method to identify a suitable order of the variables, which allows us to obtain consensus BNs having (by far) less edges than those obtained by using random orderings. Furthermore, we apply an optimization method based on the GES algorithm to remove the extra edges. As GES is a data-driven method and we have not data but a set of incoming networks, we propose to use the independences codified in the incoming networks to determine a score in order to evaluate the goodness of removing a given edge. From the experiments carried out, we observe that our heuristic is very competitive, driving the fusion process to solutions close to the optimal one.
This work has been partially funded by FEDER funds, the Spanish Goverment (AEI/MINECO) through the project TIN2016-77902-C3-1-P and the Regional Government (JCCM) by SBPLY/17/180501/000493.
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Notes
- 1.
We denote by \(pa(X_i)\) (\(pa_G(X_i)\)) the parent set of \(X_i\) in G. Analogously, we denote by \(ch(X_i)\) (\(ch_G(X_i)\)) the children set of \(X_i\). We take \(|\mathbf {V}|=n\).
- 2.
Symmetry, decomposition, weak union, contraction and intersection.
- 3.
A node with no children.
- 4.
Equivalence classes are represented by using a mixed graph structure which contains directed and undirected arcs/edges.
- 5.
It would be easy to show that GES would get the correct gold-standard DAG.
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Puerta, J.M., Aledo, J.Á., Gámez, J.A., Laborda, J.D. (2019). Structural Fusion/Aggregation of Bayesian Networks via Greedy Equivalence Search Learning Algorithm. In: Kern-Isberner, G., Ognjanović, Z. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2019. Lecture Notes in Computer Science(), vol 11726. Springer, Cham. https://doi.org/10.1007/978-3-030-29765-7_36
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