Algorithms for probabilistic inference in Bayesian networks are known to have running times that are worst-case exponential in the size of the network. For networks with a moralised graph of bounded treewidth, however, these algorithms take a time which is linear in the network's size. In this paper, we show that under the assumption of the Exponential Time Hypothesis (ETH), small treewidth of the moralised graph actually is a necessary condition for a Bayesian network to render inference efficient by an algorithm accepting arbitrary instances. We thus show that no algorithm can exist that performs inference on arbitrary Bayesian networks of unbounded treewidth in polynomial time, unless the ETH fails.
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