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What is WWB (Weiss–Weinstein bound)

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Weiss-Weinstein Bound (WWB)

Introduction

The Weiss-Weinstein Bound (WWB) is a fundamental tool in estimation theory, providing a lower bound on the mean squared error (MSE) of an estimator. It's particularly useful when dealing with Bayesian estimation problems where prior information about the parameter to be estimated is available.

Key Concepts

  • Bayesian Estimation: In this framework, the parameter to be estimated is treated as a random variable with a known prior distribution. The goal is to find an estimator that minimizes the expected posterior loss, often the mean squared error.
  • Mean Squared Error (MSE): A common performance metric for estimators, measuring the average squared difference between the estimated value and the true value.
  • Lower Bound: The WWB provides a theoretical limit on how well an estimator can perform in terms of MSE, given the available information.

The WWB Formula

The WWB is generally expressed as a complex optimization problem, but its core idea involves finding the tightest lower bound on the MSE by maximizing a specific function over a set of probability density functions (pdfs).

While the exact mathematical formulation is complex, the key intuition is that the WWB is derived by considering a class of test statistics and finding the one that maximizes the expected value of a certain function of the test statistic and the parameter to be estimated.

Properties of the WWB

  • Generality: The WWB is applicable to a wide range of estimation problems, including both continuous and discrete parameters.
  • Tightness: It often provides tighter bounds compared to other classical bounds like the Cramér-Rao Bound (CRB), especially in non-regular cases.
  • Computational Complexity: Calculating the WWB can be computationally demanding, especially for complex models.

Applications of the WWB

  • Performance Benchmarking: The WWB serves as a benchmark for evaluating the performance of different estimators.
  • Estimator Design: It can guide the design of new estimators by providing a target for performance improvement.
  • System Analysis: The WWB can be used to analyze the fundamental limits of estimation in various systems, such as communication systems, radar systems, and signal processing applications.

Comparison with Other Bounds

  • Cramér-Rao Bound (CRB): The CRB is another well-known lower bound, but it requires stronger regularity conditions on the probability density functions. The WWB is often tighter than the CRB in cases where these conditions are not met.
  • Bayesian Cramér-Rao Bound (BCRB): The BCRB is a Bayesian counterpart to the CRB, but it also has limitations in terms of applicability. The WWB can be a more flexible alternative.

Conclusion

The Weiss-Weinstein Bound is a powerful tool for analyzing the performance of estimators in Bayesian estimation problems. Its ability to provide tight lower bounds without restrictive assumptions makes it a valuable tool in various fields of engineering and science.

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