Learning and Sampling of Atomic Interventions from Observations

We study the problem of efficiently estimating the effect of an intervention on a single variable (atomic interventions) 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 $\vec{V}$ 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. Assuming that $G$ has bounded in-degree, bounded c-components ($k$), and that the observational distribution is identifiable and satisfies certain strong positivity condition, we give an algorithm that takes $m=\tilde{O}(n{ϵ}^{-2})$ samples from $P$ and $O(mn)$ time, and outputs with high probability a description of a distribution $\widehat{P}$ such that $d_{\mathrm{T}\mathrm{V}}(P_x,\widehat{P})\le ϵ$, and:

1. [Evaluation] the description can return in $O(n)$ time the probability $\widehat{P}(\vec{v})$ for any assignment $\vec{v}$ to $\vec{V}$
2. [Generation] the description can return an iid sample from $\widehat{P}$ in $O(n)$ time. We also show lower bounds for the sample complexity showing that our sample complexity has an optimal dependence on the parameters $n$ and $ϵ$, as well as if $k=1$ on the strong positivity parameter.