What is the best estimate for a random variable if we consider the square error?
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 $\E (x - \xi )^2$.
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 $\E \norm{x - \xi }^2$.