Best affine state estimate ($\arg {\min }_{a,C}\mathbb{E}\Vert x-a-Cy{\Vert }^2$)

A best affine state estimate minimizes mean squared error among estimates $a+Cy$ formed from the available observation vector $y$. With finite second moments and invertible observation covariance matrix, it isThis follows from the orthogonal projection of centered state components onto the linear span of centered observation components. For Jointly Gaussian state and observation it equals the conditional expectation; for general laws a nonlinear estimate can have lower mean squared error.