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We derive a connection between performance of estimators the performance ofthe ideal observer on related detection tasks. Specifically we show how ShannonInformation for the task of detecting a change in a parameter is related to theFisher Information and the Bayesian Fisher Information. We have previouslyshown that this Shannon Information is related via an integral transform to theMinimum Probability of Error on the same task. We then outline a circle ofrelations starting with this Minimum Probability of Error and Ensemble MeanSquared Error via the Ziv-Zakai inequality, then the Ensemble Mean Squarederror for an estimator and the Bayesian Fisher Information via the van TreesInequality, and finally the Bayesian Fisher Information and the ShannonInformation for a detection task via the work here.

The Relation Between Bayesian Fisher Information and Shannon Information for Detecting a Change in a Parameter