Learning and Testing Causal Models with Interventions
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We consider testing and learning problems on causal Bayesian networks as defined by Pearl (Pearl, 2009). Given a causal Bayesian network M on a graph with n discrete variables and bounded in-degree and bounded `confounded components', we show that O( n) interventions on an unknown causal Bayesian network X on the same graph, and Õ(n/ϵ^2) samples per intervention, suffice to efficiently distinguish whether X=M or whether there exists some intervention under which X and M are farther than ϵ in total variation distance. We also obtain sample/time/intervention efficient algorithms for: (i) testing the identity of two unknown causal Bayesian networks on the same graph; and (ii) learning a causal Bayesian network on a given graph. Although our algorithms are non-adaptive, we show that adaptivity does not help in general: Ω( n) interventions are necessary for testing the identity of two unknown causal Bayesian networks on the same graph, even adaptively. Our algorithms are enabled by a new subadditivity inequality for the squared Hellinger distance between two causal Bayesian networks.
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