Scalar Gaussian Kalman recursion ($K_t=F{R}_t/(F^2{R}_t+V)$)

For the state-space model $S_t=G{S}_{t-1}+w_t$, $X_t=F{S}_t+v_t$, assume independent Gaussian process noise and measurement noise of variances $W,V$, independent of the prior state and previous observations. A normal distribution for the filtered state, with mean $m$ and variance $P$, predicts mean $Gm$ and variance $R=G^{2}P+W$. Here $R$ denotes predicted state variance, rather than the observation-noise covariance notation used in the vector equations above. Gaussian conditional expectation then gives posterior mean $Gm+K(X-FGm)$ and posterior variance $RV/(F^{2}R+V)$, where the Kalman gain is $K=FR/(F^{2}R+V)$.

Mere weak white noise assumptions do not imply normal posterior distributions: independent process noise with a Rademacher distribution makes the next state a mixture of two shifted normal distributions. The Kalman filter can still describe a best affine state estimate under appropriate second-order assumptions, but that is distinct from an exact Gaussian conditional law.

For the state-space model with both scalar coefficients equal to one, ${\alpha }_t=(P_{t-1}+W)/(P_{t-1}+W+V)$ and $P_t={\alpha }_{t}V$. If the variance converges and $c=W/V$, its limiting gain satisfies ${\alpha }^2+c\alpha -c=0$. The nonnegative root lies below one for finite $c>0$; it tends to zero with $c$ and to one as $c$ grows. Thus noisier state evolution assigns more weight to the latest observation.