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REVIEWER 5: Yes, clarifying that we assume chordality is useful, and will revise the title, abstract and elsewhere to emphasize this assumption. REVIEWER 6: The reviewer's summary of the proof of Lemma 4 about the balancing condition is accurate. We may have been a bit pedantic in spelling out the details of the proof, but on the other hand, simply saying that the balancing condition "obviously" holds because of the running intersection property would not be very informative either, and we would rather err on the side of giving too much details rather than too little. The standard Bayesian approach we use for model learning is statistically consistent for choosing the correct dimensionality, since prior distribution assigned to model parameters acts as a regularizer. This property is so widely established in the literature that we did not consider it to be necessary to emphasize the aspect in the paper.