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 ($tll N$). 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.
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