Efficient inference of interventional distributions

07/25/2021

∙

by   Arnab Bhattacharyya, et al.

∙

0

∙

share

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⊆𝐕, and setting x to 𝐗, let P_ x(𝐘) denote 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), gave an exact characterization of the class of causal graphs for which the interventional distribution P_ x(𝐘) can be uniquely determined. We give the first efficient version of the Shpitser-Pearl algorithm. In particular, under natural assumptions, we give a polynomial-time algorithm that on input a causal graph G on observable variables 𝐕, a setting x of a set 𝐗⊆𝐕 of bounded size, outputs succinct descriptions of both an evaluator and a generator for a distribution P̂ that is ε-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 set, there is no efficient algorithm that outputs an evaluator of a distribution that is ε-close to P_ x(𝐘) unless all problems that have statistical zero-knowledge proofs, including the Graph Isomorphism problem, have efficient randomized algorithms.