Additional measurements decrease known-error covariance ($(F+G{)}^{-1}⪯F^{-1},F\succ 0,G⪰0$)

Independent observations with known errors add a positive-semidefinite Fisher information matrix $G$ to the existing information $F$. Writing $F+G=F^{1/2}(I+F^{-1/2}G{F}^{-1/2})F^{1/2}$ 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.