Minimum Mean Squared Error Estimates

Why

What is the best estimate for a random variable if we consider the square error?

Definition

Let $(\Omega , \mathcal{A} , \mathbfsf{P} )$ be a probability space and $x: \Omega \to \R $ a random variable. A minimum mean squared error estimate or MMSE estimate or least square estimate is a value $\xi \in \R $ which minimizes $\E (x - \xi )^2$.

There is a unique MMSE estimate and it is given by $\E (x)$.

Vector case

Let $(\Omega , \mathcal{A} , \mathbfsf{P} )$ be a probability space and $y: \Omega \to \R ^n$ a random variable.¹

A minimum mean squared error estimator or MMSE estimator or least square estimator is a value $\xi \in \R $ which minimizes $\E \norm{x - \xi }^2$.

There is a unique MMSE estimator and it is given by $\E (y)$.

Future editions might collapse this into the previous case. ↩︎