  Paired estimation of change in parameters of interest over a population plays
a central role in several application domains including those in the social
sciences, epidemiology, medicine and biology. In these domains, the size of the
population under study is often very large, however, the number of observations
available per individual in the population is very small (\emph{sparse
observations}) which makes the problem challenging. Consider the setting with
$N$ independent individuals, each with unknown parameters $(p_i, q_i)$ drawn
from some unknown distribution on $[0, 1]^2$. We observe $X_i \sim
\text{Bin}(t, p_i)$ before an event and $Y_i \sim \text{Bin}(t, q_i)$ after the
event. Provided these paired observations, $\{(X_i, Y_i) \}_{i=1}^N$, our goal
is to accurately estimate the \emph{distribution of the change in parameters},
$\delta_i := q_i - p_i$, over the population and properties of interest like
the \emph{$\ell_1$-magnitude of the change} with sparse observations ($t\ll
N$). We provide \emph{information theoretic lower bounds} on the error in
estimating the distribution of change and the $\ell_1$-magnitude of change.
Furthermore, we show that the following two step procedure achieves the optimal
error bounds: first, estimate the full joint distribution of the paired
parameters using the maximum likelihood estimator (MLE) and then estimate the
distribution of change and the $\ell_1$-magnitude of change using the joint
MLE. Notably, and perhaps surprisingly, these error bounds are of the same
order as the minimax optimal error bounds for learning the \emph{full} joint
distribution itself (in Wasserstein-1 distance); in other words, estimating the
magnitude of the change of parameters over the population is, in a minimax
sense, as difficult as estimating the full joint distribution itself.
