  We examine the extent to which sublinear-sample property testing and
estimation applies to settings where samples are independently but not
identically distributed. Specifically, we consider the following distributional
property testing framework: Suppose there is a set of distributions over a
discrete support of size $k$, $\textbf{p}_1, \textbf{p}_2,\ldots,\textbf{p}_T$,
and we obtain $c$ independent draws from each distribution. Suppose the goal is
to learn or test a property of the average distribution,
$\textbf{p}_{\mathrm{avg}}$. This setup models a number of important practical
settings where the individual distributions correspond to heterogeneous
entities -- either individuals, chronologically distinct time periods,
spatially separated data sources, etc. From a learning standpoint, even with
$c=1$ samples from each distribution, $\Theta(k/\varepsilon^2)$ samples are
necessary and sufficient to learn $\textbf{p}_{\mathrm{avg}}$ to within error
$\varepsilon$ in TV distance. To test uniformity or identity -- distinguishing
the case that $\textbf{p}_{\mathrm{avg}}$ is equal to some reference
distribution, versus has $\ell_1$ distance at least $\varepsilon$ from the
reference distribution, we show that a linear number of samples in $k$ is
necessary given $c=1$ samples from each distribution. In contrast, for $c \ge
2$, we recover the usual sublinear sample testing of the i.i.d. setting: we
show that $O(\sqrt{k}/\varepsilon^2 + 1/\varepsilon^4)$ samples are sufficient,
matching the optimal sample complexity in the i.i.d. case in the regime where
$\varepsilon \ge k^{-1/4}$. Additionally, we show that in the $c=2$ case, there
is a constant $\rho > 0$ such that even in the linear regime with $\rho k$
samples, no tester that considers the multiset of samples (ignoring which
samples were drawn from the same $\textbf{p}_i$) can perform uniformity
testing.
