minimummeansquareerror

The Minimum Mean Square Error (MMSE) is a criterion used in estimation theory to find the "best" estimate of an unknown quantity. It is defined as the estimator that minimizes the expected value of the squared difference between the estimated value and the true value. Mathematically, if $\hat{x}$ is an estimator for a random variable $x$, the MMSE estimator $\hat{x}_{MMSE}$ is chosen such that $E[(\hat{x} - x)^2]$ is minimized.

This criterion is widely used in various fields, including signal processing, statistics, and machine learning, for tasks such as system identification, noise reduction, and parameter estimation. The MMSE estimator provides a theoretically optimal solution in terms of minimizing the average squared error.

The calculation of the MMSE estimator often depends on the statistical properties of the random variable and any observed data. If the joint probability distribution of the true value and the observed data is known, the MMSE estimator can be derived as the conditional expectation of the true value given the observed data. However, in many practical scenarios, this distribution is unknown, and approximations or alternative estimation criteria, such as the Maximum Likelihood Estimation (MLE) or Minimum Variance Unbiased (MVU) estimation, are employed. The MVU estimator, when it exists, is the MMSE estimator that is also unbiased.