Relation between Bayesian Fisher information and Shannon information for detecting a change in a parameter

We derive a connection between the performance of statistical estimators and the performance of the ideal observer on related detection tasks. Specifically, we show how the task-specific Shannon information for the task of detecting a change in a parameter is related to the Fisher information and to the Bayesian Fisher information. We have previously shown that this Shannon information is related via an integral transform to the minimum probability of error on the same task. We then outline a circle of relations starting with this minimum probability of error and ensemble mean squared error for an estimator via the Ziv-Zakai inequality, then the ensemble mean squared error and the Bayesian Fisher information via the van Trees inequality, and finally the Bayesian Fisher information and the Shannon information for a detection task via the work presented here.