ThmDex – An index of mathematical definitions, results, and conjectures.
Bias-variance partition of mean square error for a random real column matrix
Formulation 0
Let $X \in \text{Random}(\mathbb{R}^{N \times 1})$ be a D5210: Random real column matrix such that
(i) \begin{equation} \mathbb{E} |X|^2 < \infty \end{equation}
Let $a \in \mathbb{R}^{N \times 1}$ be a D5200: Real column matrix.
Then \begin{equation} \mathbb{E} |X - a|^2 = \mathbb{E} |X - \mathbb{E} X|^2 - |\mathbb{E} X - a|^2 \end{equation}
Formulation 3
Let $X \in \text{Random}(\mathbb{R}^{N \times 1})$ be a D5210: Random real column matrix such that
(i) \begin{equation} \mathbb{E} \Vert X \Vert^2_2 < \infty \end{equation}
Let $a \in \mathbb{R}^{N \times 1}$ be a D5200: Real column matrix.
Then \begin{equation} \mathbb{E} \Vert X - a \Vert^2_2 = \mathbb{E} \Vert X - \mathbb{E} X \Vert^2_2 - \Vert \mathbb{E} X - a \Vert^2_2 \end{equation}
Proofs
Proof 0
Let $X \in \text{Random}(\mathbb{R}^{N \times 1})$ be a D5210: Random real column matrix such that
(i) \begin{equation} \mathbb{E} |X|^2 < \infty \end{equation}
Let $a \in \mathbb{R}^{N \times 1}$ be a D5200: Real column matrix.
This result is a particular case of R4970: . $\square$