Probability of error for detecting a change in a parameter and Bayesian Fisher information

Area of Science:

• Information Theory
• Statistical Inference
• Estimation Theory

Background:

• The van Trees inequality links estimator error to Bayesian Fisher information.
• The Ziv-Zakai inequality links estimator error to minimum probability of error in parameter change detection.

Purpose of the Study:

• To derive a novel inequality completing the circle between minimum probability of error and Bayesian Fisher information.
• To explore this relationship for both scalar and vector parameters.

Main Methods:

• Derivation of a new inequality relating minimum probability of error to Bayesian Fisher information.
• Analysis of the role of posterior probability distribution's total variation.

Main Results:

• A new inequality is established, connecting minimum probability of error to Bayesian Fisher information.
• The total variation of the posterior probability distribution is identified as a key intermediary, potentially easier to compute.

Conclusions:

• The derived inequality provides a unified framework linking key concepts in statistical inference.
• The total variation offers a potentially more accessible metric for analysis in parameter estimation and detection problems.