  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.
