Algorithms for learning Bayesian networks from data have two components: a scoring metric and a search procedure. The scoring metric computes a score re ecting the goodness-of-t of the structure to the data. The search procedure tries to identify network structures with high scores. Heckerman et al. (1994) introduced a Bayesian metric, called the BDe metric, that computes the relative posterior probability of a network structure given data. They show that the metric has a property desireable

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... inferring causal structure from data. In this paper, we show that the problem of deciding whether there is a Bayesian network|among those where each node has at most k parents|that has a relative posterior probability greater than a given constant is NP-complete, when the BDe metric is used.