Fisher Information and Mutual Information Constraints
share
We consider the processing of statistical samples X∼ P_θ by a channel p(y|x), and characterize how the statistical information from the samples for estimating the parameter θ∈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 θ 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.
READ FULL TEXT