What is MMSE minimum mean-squared error

Technical details of Minimum Mean-Squared Error (MMSE) estimation.

1. Motivation:
   • MMSE estimation is a powerful technique used in statistics, signal processing, and other fields. It aims to estimate an unknown signal based on noisy measurements.
   • The primary goal of MMSE is to minimize the mean-squared error (MSE) between the estimated signal and the true signal.
   • In the Bayesian setting, MMSE specifically refers to estimation with a quadratic loss function.
2. Bayesian Approach:
   • Unlike non-Bayesian approaches (such as the minimum-variance unbiased estimator), where we assume nothing about the parameter in advance, the Bayesian approach incorporates prior information.
   • In practical scenarios, we often have some prior knowledge about the parameter to be estimated. This could be a range of possible values, an old estimate, or statistical properties of a random signal.
   • The Bayesian approach captures this prior information using a prior probability density function for the parameters.
   • As more observations become available, the Bayesian estimator updates its estimates based on Bayes’ theorem, leading to better posterior estimates.
3. Definition:
   • Let’s consider a hidden random vector variable X (the true signal) and a known random vector variable Y (the noisy measurement or observation). These vectors may not necessarily be of the same dimension.
   • An estimator of X is any function of the measurement Y.
   • The estimation error vector is given by e = X – Y, and its mean squared error (MSE) is calculated as the trace of the error covariance matrix.
4. Linear MMSE Estimation:
   • Linear MMSE estimators are popular due to their simplicity, versatility, and ease of calculation.
   • The MMSE estimator seeks to minimize the MSE by finding an optimal linear combination of the noisy measurements.
   • The Wiener–Kolmogorov filter and the Kalman filter are well-known examples of linear MMSE estimators.
5. Mathematical Formulation:
   • Let θ represent the parameter to be estimated.
   • The MMSE estimator, denoted as g(Y), is given by the posterior mean of θ: [ g(Y) = \mathbb{E}[\theta | Y] ]
   • Calculating the exact posterior mean can be cumbersome, so we often constrain the form of the MMSE estimator to a certain class of functions.
6. Applications:
7. Signal Processing: MMSE is widely used in denoising, channel equalization, and parameter estimation.