= Gaussian filtering with exact observations
{c}
{title2=$R_t=GP_{t-1}G^\top+Q,\quad V_t=FR_tF^\top$}
In a linear <Gaussian> <state-space model> with <process noise> <covariance> $Q$ and exact scalar observation $X_t=FS_t$, the predictive state <covariance> includes $Q$ even though there is no separate <measurement noise>. Here $R_t=GP_{t-1}G^\top+Q$ denotes predicted state <covariance>; it differs from the measurement-noise notation in the general <Kalman filter> equations. The observation <innovation> <variance> is $V_t=FGP_{t-1}G^\top F^\top+FQF^\top$. Omitting the second term incorrectly removes the current <process noise>. If $V_t>0$, <Gaussian conditional expectation> gives the <posterior covariance matrix> $P_t=R_t-R_tF^\top FR_t/V_t$. This can be a <singular matrix> because the observed <linear combination> is then known exactly.
For a concrete example with two state components that are <dimensionless quantities>, take predicted mean $(10,0)^\top$, predicted <covariance matrix>
$$
R=\begin{pmatrix}4&2\\2&9\end{pmatrix},
$$
and exact observation of the first component, $F=(1,0)$, $X=12$. The <innovation> is $2$, its <variance> is $4$, and the <Kalman gain> is $(1,1/2)^\top$. The <posterior mean> and <posterior covariance matrix> are therefore
$$
m=\begin{pmatrix}12\\1\end{pmatrix},\qquad
P=R-\frac14\begin{pmatrix}4\\2\end{pmatrix}\begin{pmatrix}4&2\end{pmatrix}
=\begin{pmatrix}0&0\\0&8\end{pmatrix}.
$$
The observed component now has zero <variance>. The second component remains uncertain, but its <posterior mean> changes and its <variance> falls from $9$ to $8$ because its prediction error had nonzero <correlation> with that of the first. This is the mechanism by which measurements of one component can inform another.
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