Probability of Error for Detecting a Change in a Parameter, Total Variation of the Posterior Distribution, and Bayesian Fisher Information

The van Trees inequality relates the Ensemble Mean Squared Error of an estimator to a Bayesian version of the Fisher Information. The Ziv-Zakai inequality relates the Ensemble Mean Squared Error of an estimator to the Minimum Probability of Error for the task of detecting a change in the parameter. In this work we complete this circle by deriving an inequality that relates this Minimum Probability of Error to the Bayesian version of the Fisher Information. We discuss this result for both scalar and vector parameters. In the process we discover that an important intermediary in the calculation is the Total Variation of the posterior probability distribiution function for the parameter given the data. This total variation is of interest in its own right since it may be easier to compute than the other figures of merit discussed here. Examples are provided to show that the inequality derived here is sharp.