  We consider the processing of statistical samples $X\sim P_\theta$ by a
channel $p(y|x)$, and characterize how the statistical information from the
samples for estimating the parameter $\theta\in\mathbb{R}^d$ can scale with the
mutual information or capacity of the channel. We show that if the statistical
model has a sub-Gaussian score function, then the trace of the Fisher
information matrix for estimating $\theta$ from $Y$ can scale at most linearly
with the mutual information between $X$ and $Y$. We apply this result to obtain
minimax lower bounds in distributed statistical estimation problems, and obtain
a tight preconstant for Gaussian mean estimation. We then show how our Fisher
information bound can also imply mutual information or Jensen-Shannon
divergence based distributed strong data processing inequalities.
