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In a linear Gaussian state-space model with process noise covariance $Q$ and exact scalar observation $X_t=F{S}_t$, the predictive state covariance includes $Q$ even though there is no separate measurement noise. Here $R_t=G{P}_{t-1}{G}^{\top }+Q$ denotes predicted state covariance; it differs from the measurement-noise notation in the general Kalman filter equations. The observation innovation variance is $V_t=FG{P}_{t-1}{G}^{\top }{F}^{\top }+FQ{F}^{\top }$. Omitting the second term incorrectly removes the current process noise. If $V_t>0$, Gaussian conditional expectation gives the posterior covariance matrix $P_t=R_t-R_t{F}^{\top }F{R}_t/V_t$. 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 matrixand 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 thereforeThe 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.

Consider a local-level state-space model $x_t=x_{t-1}+w_t$, $y_t=x_t+v_t$, with mutually independent centered Gaussian noises of variances $Q=1/4$ and $R=1$. Start with $x_0\sim N(10,4)$, independently of these noises. Before the first observation, the Kalman prediction step gives $m_1^{-}=10$, $P_1^{-}=17/4$. If $y_1=14$, the Kalman gain is $17/21$, and the posterior mean and posterior variance becomeThe next Kalman prediction step keeps the mean $m_2^{-}=278/21$ but adds fresh process noise variance, giving $P_2^{-}=89/84$. If $y_2=12$, thenThus new uncertainty is introduced at each transition and reduced by each measurement. It does not shrink to zero merely because many measurements have arrived: the state continues to change.

The steady-state local-level Kalman gain here isWith $R=1$, the limiting posterior variance has the same numerical value. By contrast, the static-state model $Q=0$ with $P_0=4$ gives $P_t=1/(t+1/4)$ and $K_t=1/(t+1/4)$. That decreasing gain is appropriate for a state that truly remains constant. Applied to the changing random walk, it builds confidence in the wrong model. The choice of process noise is therefore a statement about state evolution, rather than just an instruction to make a plot look smoother.

The Kalman prediction step advances the previous posterior mean and posterior covariance matrix through the state transition:The minus superscript marks information before observing $y_t$. The transpose and matrix multiplication propagate the old uncertainty; $Q_t$ adds the uncertainty of the new process noise. A known control input changes the mean but contributes no additional variance. If the control input itself is uncertain, that uncertainty needs its own model.

Process noise is the random contribution to a state transition in a state-space model. It represents uncertainty in how the state evolves. Measurement noise instead affects the observed value of the state. In the local-level state-space model $x_t=x_{t-1}+w_t$, $y_t=x_t+v_t$, the first disturbance changes the state itself; the second changes its measurement.

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.

State prediction estimates a future state of a dynamical system using observations already available. A state-space model supplies the transition law. Unlike causal state filtering at the next time, the prediction has not yet incorporated that next observation; a Kalman prediction step therefore includes uncertainty from new process noise.