# Efficiently Learning and Sampling Interventional Distributions from Observations

We study the problem of efficiently estimating the effect of an intervention on a single variable using observational samples in a causal Bayesian network. Our goal is to give algorithms that are efficient in both time and sample complexity in a non-parametric setting. Tian and Pearl (AAAI `02) have exactly characterized the class of causal graphs for which causal effects of atomic interventions can be identified from observational data. We make their result quantitative. Suppose P is a causal model on a set V of n observable variables with respect to a given causal graph G with observable distribution P. Let P_x denote the interventional distribution over the observables with respect to an intervention of a designated variable X with x. We show that assuming that G has bounded in-degree, bounded c-components, and that the observational distribution is identifiable and satisfies certain strong positivity condition: 1. [Evaluation] There is an algorithm that outputs with probability 2/3 an evaluator for a distribution P' that satisfies d_tv(P_x, P') ≤ϵ using m=Õ(nϵ^-2) samples from P and O(mn) time. The evaluator can return in O(n) time the probability P'(v) for any assignment v to V. 2. [Generation] There is an algorithm that outputs with probability 2/3 a sampler for a distribution P̂ that satisfies d_tv(P_x, P̂) ≤ϵ using m=Õ(nϵ^-2) samples from P and O(mn) time. The sampler returns an iid sample from P̂ with probability 1-δ in O(nϵ^-1logδ^-1) time. We extend our techniques to estimate marginals P_x|_Y over a given Y ⊂ V of interest. We also show lower bounds for the sample complexity showing that our sample complexity has optimal dependence on the parameters n and ϵ as well as the strong positivity parameter.

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