Kalman gain ($K_t=P_t^{-}{H}_t^{\top }{S}_t^{-1}$)

The Kalman gain maps an observation innovation into a state correction:For a scalar state observed directly, it reduces to $K=P^{-}/(P^{-}+R)$. With positive variances, $0<K<1$, and the Kalman measurement update is the weighted mean $(1-K)m^{-}+Ky$. That interpretation belongs to the scalar unit-coefficient case; a general Kalman gain is a rectangular matrix, not a collection of weights necessarily between zero and one.