Efficient Interventional Distribution Learning in the PAC Framework

We consider the problem of efficiently inferring interventional distributions in a causal Bayesian network from a finite number of observations. Let P be a causal model on a set V of observable variables on a given causal graph G. For sets X, Y subset of V, and setting x to X, P-x (Y) denotes the interventional distribution on Y with respect to an intervention x to variables X. Shpitser and Pearl (AAAI 2006), building on the work of Tian and Pearl (AAAI 2001), proved that the ID algorithm is sound and complete for recovering P-x (Y) from observations. We give the first provably efficient version of the ID algorithm. In particular, under natural assumptions, we give a polynomialtime algorithm that on input a causal graph G on observable variables V, a setting x of a set X subset of V of bounded size, outputs succinct descriptions of both an evaluator and a generator for a distribution (P) over cap that is epsilon-close (in total variation distance) to P-x (Y) where Y = V \ X, if P-x (Y) is identifiable. We also show that when Y is an arbitrary subset of V \ X, there is no efficient algorithm that outputs an evaluator of a distribution that is E-close to P-x (Y) unless all problems that have statistical zero-knowledge proofs, including the Graph Isomorphism problem, have efficient randomized algorithms.