Technical details of the Minimum Mean Square Error (MMSE) estimator.
- Motivation:
- The MMSE estimator is a statistical method used for estimating an unknown parameter based on observed data.
- It aims to minimize the mean square error (MSE), which is a common measure of the quality of an estimator.
- In the Bayesian setting, the term “MMSE” specifically refers to estimation with a quadratic loss function.
- Bayesian Approach:
- Unlike non-Bayesian approaches (such as the minimum-variance unbiased estimator), the Bayesian approach incorporates prior information about the parameter to be estimated.
- We often have some prior knowledge about the parameter, such as its possible range or an old estimate.
- The Bayesian estimator treats the parameter itself as a random variable, allowing us to make better estimates as more observations become available.
- Definition:
- Let’s consider a hidden random vector variable denoted as X, and a known random vector variable (the measurement or observation) denoted as Y.
- An estimator of X is any function of the measurement Y.
- The estimation error vector is given by e = X – Y, and its MSE is calculated as the trace of the error covariance matrix.
- Linear MMSE Estimators:
- Linear MMSE estimators are popular due to their simplicity and versatility.
- They are widely used in signal processing and communication systems.
- The linear MMSE estimator seeks to find the best linear combination of the observed data to estimate the hidden variable.
- Examples of linear MMSE estimators include the Wiener–Kolmogorov filter and the Kalman filter.
- Mathematical Formulation:
- Suppose we want to estimate a parameter vector θ based on the observation vector y.
- The MMSE estimator is given by the posterior mean of θ: [ \hat{\theta}_{\text{MMSE}} = \mathbb{E}(\theta | y) ]
- Calculating the exact posterior mean can be cumbersome, so we often constrain the form of the estimator to a certain class of functions.
- Application in Communication Systems:
- In digital communication systems, the MMSE equalizer is used to mitigate the effects of inter-symbol interference (ISI) and noise.
- The MMSE equalizer minimizes the mean squared error between the transmitted signal and the equalized received signal.
