= Scalar Gaussian Kalman recursion
{title2=$K_t=FR_t/(F^2R_t+V)$}
For the <state-space model> $S_t=GS_{t-1}+w_t$, $X_t=FS_t+v_t$, 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> $R=G^2P+W$. 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> $RV/(F^2R+V)$, where the <Kalman gain> is $K=FR/(F^2R+V)$.
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.
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