  We consider the basic statistical problem of detecting truncation of the
uniform distribution on the Boolean hypercube by juntas. More concretely, we
give upper and lower bounds on the problem of distinguishing between i.i.d.
sample access to either (a) the uniform distribution over $\{0,1\}^n$, or (b)
the uniform distribution over $\{0,1\}^n$ conditioned on the satisfying
assignments of a $k$-junta $f: \{0,1\}^n\to\{0,1\}$.
  We show that (up to constant factors) $\min\{2^k + \log{n\choose k},
{2^{k/2}\log^{1/2}{n\choose k}}\}$ samples suffice for this task and also show
that a $\log{n\choose k}$ dependence on sample complexity is unavoidable. Our
results suggest that testing junta truncation requires learning the set of
relevant variables of the junta.
