  We derive a connection between performance of estimators the performance of
the ideal observer on related detection tasks. Specifically we show how Shannon
Information for the task of detecting a change in a parameter is related to the
Fisher Information and 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 via the Ziv-Zakai inequality, then the Ensemble Mean Squared
error for an estimator 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 here.
