In the machine learning community, the Bayesian scoring criterion is widely used for model selection problems. One of the fundamental theoretical properties justifying the usage of the Bayesian scoring criterion is its consistency. In this paper we refine this property for the case of binomial Bayesian network models. As a by-product of our derivations we establish strong consistency and obtain the law of iterated logarithm for the Bayesian scoring criterion. © 2009 Nikolai Slobodianik, Dmitry Zaporozhets and Neal Madras.