  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 $m=\tilde{O}(n\epsilon^{-2})$ samples from $P$ and $O(mn)$
time, and outputs with high probability a description of a distribution
$\hat{P}$ such that $d_{\mathrm{TV}}(P_x, \hat{P}) \leq \epsilon$, and:
  1. [Evaluation] the description can return in $O(n)$ time the probability
$\hat{P}(\vec{v})$ for any assignment $\vec{v}$ to $\vec{V}$
  2. [Generation] the description can return an iid sample from $\hat{P}$ 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 $\epsilon$, as
well as if $k=1$ on the strong positivity parameter.
