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I'm recently reading a paper about Scoring Mechanisms for Bayesian Networks.

For the BDeu score, it appears that the maximum possible score of BDeu for Bayesian Network structure learning is zero.

Does it mean that the best network is always the empty network?

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Does it mean that the best network is always the empty network?

No.

The empty DAG (i.e. the DAG with no arcs) implies the factorization is simply $$P(X_1, \dots, X_n) = P(X_1) \dots P(X_n),$$ which is telling you all variables are independent.

This will be the best network (DAG) if all the variables are actually independent (Remember that independence for a dataset entails a certain tolerance), because the penalization coefficient will make the score lower as any arc is added.

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