= Additional measurements decrease known-error covariance
{title2=$(F+G)^{-1}\preceq F^{-1},\quad F\succ0,\ G\succeq0$}
Independent observations with known errors add a positive-semidefinite <Fisher information matrix> $G$ to the existing information $F$. Writing $F+G=F^{1/2}(I+F^{-1/2}GF^{-1/2})F^{1/2}$ shows its inverse covariance cannot increase in any parameter direction. Improvement can be zero along an uninformative direction. This statement does not assume that data-dependent residual <variance> estimates or model <systematic errors> must decrease.
Back to article page
