... function¹

Some authors also use the term scoring metric.

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... nodes²

For reasons of simplicity, the set of nodes which are in one-to-one correspondence with the variables in ${\mathcal{U}}$ will also be denoted by ${\mathcal{U}}$.

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...equivalent³

Several authors also use the term independence equivalent, and reserve the term distribution equivalent (wrt some family of distributions ${\cal F}$) for the more restricted case where the two DAGs can represent the same probability distributions. In the common situation where all the variables are discrete and ${\cal F}$ is the family of unrestricted multinomial distributions, these two concepts of equivalence coincide.

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... classes⁴

Note that the operators of addition and reversal of an arc in the DAG space also need a consistency check, but in this case we simply test the absence of directed cycles.

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... component⁵

A chain component of $G$ is any connected component of the undirected graph obtained from $G$ by removing all the arcs.

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...arb_stats⁶

This tree could be organized differently in order to improve the efficiency in the location of the current state. However, this particular tree was selected so as to clarify the presentation.

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...Chickering02⁷

This work appeared after the original submission of this paper.

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