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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.

The Kalman measurement update incorporates the new observation by Gaussian conditional expectation:The posterior covariance matrix subtracts a positive semidefinite matrix, reflecting the information supplied by the measurement. Even if the innovation is zero and the mean does not move, the uncertainty can decrease. In this linear Gaussian model the posterior covariance matrix depends on the observation geometry and its uncertainty, rather than on the numerical observed value.

The derivation is one application of Gaussian conditional expectation per measurement. Before observing $y_t$, the state and observation have joint covariance matrixConditioning the upper block on the lower block gives exactly the two update equations. The upper-right block explains the Kalman gain: a measured component corrects a state component only to the extent that their prediction errors have covariance. The measurement can therefore improve an unobserved velocity when a state-space model correlates its error with an observed position.

Let $M$ be the linear subspace of $L^2(P)$ consisting of classes admitting $\mathcal{G}$-measurable representatives. It is closed. Indeed, if $Y_n\in M$ converges in $L^2$ to $Y$, choose a subsequence converging almost surely: take $\Vert Y_{n_j}-Y{\Vert }_2^2\le 2^{-3j}$ and apply Markov inequality and the Borel-Cantelli lemma to errors exceeding $2^{-j}$. The pointwise limit of its $\mathcal{G}$-measurable representatives, set to zero wherever the finite limit does not exist, is $\mathcal{G}$-measurable and equals $Y$ almost surely. This also works when $\mathcal{G}$ is not complete.

Here is a proof of the orthogonal projection construction using only completeness. Put $d={\inf }_{W\in M}\Vert X-W{\Vert }_2$ and take $W_n$ approaching this infimum. The parallelogram law givesCompleteness and closedness give a minimizer $Y\in M$. For every $Z\in M$, expand $\Vert X-Y-tZ{\Vert }_2^2$ for real $t$. Its minimum at zero impliesThis $Y$ is unique almost surely, because the difference of two possible choices belongs to $M$ and is orthogonal to itself. It is the conditional expectation $E(X\mid \mathcal{G})$, as also follows by testing indicators of events in $\mathcal{G}$.

For the jointly Gaussian pair, if $u>0$, setThe residual $R=X-Y$ has mean zero and covariance $\mathrm{Cov}(R,G)=v-(v/u)u=0$. Since $(R,G)$ is jointly Gaussian, its characteristic function factors when this covariance is zero, so $R$ and $G$ are independent. Hence $E(RZ)=E(R)E(Z)=0$ for every square-integrable function $Z$ of $G$; independence first proves this for bounded functions, and truncation plus Cauchy-Schwarz inequality gives the general case. This proves the Gaussian conditional expectation formula, including a zero residual variance. If $u=0$, $G=\mu$ almost surely and the covariance inequality gives $v=0$; $Y=\nu$ is the answer.

Put $S=X_1+X_2$. By symmetry,Their sum is $S$, which is measurable with respect to $\mathcal{G}=\sigma (S)$, soThereforeThis also follows from the Gaussian conditional expectation formula because $\mathrm{Cov}(X_1,S)/\mathrm{Var}(S)=1/2$.

If $t\le s$, then $\mathbb{E}(X_t\mid {\mathcal{F}}_s)=X_t$. If $s<t\le 1$, Gaussian conditional expectation for the Brownian bridge between $(s,X_s)$ and $(1,X_1)$ gives

The marginal predictor density isGaussian conditional expectation givesIntegrating this conditional distribution through the response model yields

When $\tau \gg {\sigma }_x$, $\kappa \to 1$, $v\to {\sigma }_x^2$, and $A\sim {\tau }^2$. HencewhereThe approximate maximum-likelihood estimators are

From part (i),The conditional-normal formula therefore givesConsequently the Gaussian conditional expectation is

By Gaussian conditional expectation,Multiply by $U^{\prime }(X_0)$, take expectations, and use part (c):Substitution into part (d) gives

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.

Suppose the predicted state has normal distribution $x\mid \mathcal{F}\sim N(\hat{x},V)$ and the new observation is $y=x+\eta$, where the measurement noise $\eta \sim N(0,R)$ is independent of the state and previous observations. Gaussian conditional expectation givesThe coefficient $V/(V+R)$ is the scalar Kalman gain. The posterior mean is a weighted mean of the predicted state and the new observation, with the more precise quantity receiving more weight. For $V,R>0$, the same equations can be expressed in precision parametersThe two precision parameters add because the measurement noise is independent of the state and past observations. Completing the square in the product of the prior density and the likelihood function gives this form directly. This is a Gaussian conditional expectation for a fixed unknown state; a moving state additionally requires a Kalman prediction step.

Let $X=A+\epsilon$ and $S=A+\eta$, where latent score $A$ has normal distribution with mean $\mu$ and variance ${\sigma }_A^2$, and the centred normal errors are independent of each other and $A$. Since $\mathrm{Cov}(S,X)={\sigma }_A^2$ and $\mathrm{Var}(X)={\sigma }_A^2+{\sigma }_{\epsilon }^2$, Gaussian conditional expectation gives $\mathbb{E}[S\mid X=x]=\mu +\lambda (x-\mu )$ with $\lambda ={\sigma }_A^2/({\sigma }_A^2+{\sigma }_{\epsilon }^2)$. Subtracting $x$ proves the formula. For $0<\lambda <1$, selecting scores below the mean creates positive expected self-assessment errors relative to those observed scores. This is a mathematical possibility, not an empirical estimate of any particular study's contribution from measurement error.