Exact linear-Gaussian Kalman filtering

Write $\mu =\mathcal{L}(Z)$. For every $h$, the product $\mathrm{\Lambda }(h,\cdot )=\kappa (h,\cdot )\otimes \mu$ is a probability measure by [F2] and [F3]. For each $D\in \mathcal{E}\otimes \mathcal{W}$, the section $x\mapsto \mu (D_x)$ is measurable by [F4] applied to the constant probability kernel $\mu$ and the indicator of $D$. The iterated-section formula in [F3] gives $\mathrm{\Lambda }(h,D)={\int }_E\mu (D_x)\kappa (h,dx)$, which is measurable in $h$ by [F4] applied to $\kappa$. Thus $\mathrm{\Lambda }$ is a probability kernel.

Fix $C\in \sigma (Y)$ and a measurable rectangle $A\times B$. The conditional-law identity [F1], followed by independence [F6]–[F7], gives ${\int }_C\mathrm{\Lambda }(Y,A\times B)d\mathbb{P}=\mu (B){\int }_C\kappa (Y,A)d\mathbb{P}=\mu (B)\mathbb{P}(C\cap \{X\in A\})=\mathbb{P}(C\cap \{X\in A,Z\in B\})$. For this fixed $C$, both sides define finite measures on $\mathcal{E}\otimes \mathcal{W}$. They agree on the rectangle pi-system, which generates that sigma-algebra by [F5]; [F9] extends the equality to every measurable target set. Therefore $\mathrm{\Lambda }(Y,\cdot )$ is a regular conditional distribution of $(X,Z)$ given $\sigma (Y)$ by [F8].

For measurable $\mathrm{\Phi }$, every section of ${\mathrm{\Lambda }}^{\mathrm{\Phi }}$ is the pushforward of the probability $\mathrm{\Lambda }(h,\cdot )$, and its evaluations are measurable because ${\mathrm{\Phi }}^{-1}(C)$ is measurable and $\mathrm{\Lambda }$ is a probability kernel. Applying the conditional identity from step 1.2 to ${\mathrm{\Phi }}^{-1}(C)$ proves that ${\mathrm{\Lambda }}^{\mathrm{\Phi }}(Y,\cdot )$ is a regular conditional distribution of $\mathrm{\Phi }(X,Z)$ given $\sigma (Y)$. Hence it is the required conditional law.

By [F2], the stack $(U,q,r)$ is jointly Gaussian with mean $(m,0,0)$ and covariance $\mathrm{diag}(P,Q,R)$. For any $a\in {\mathbb{R}}^n$ and $b\in {\mathbb{R}}^p$, the scalar $a^{T}X+b^{T}Y$ is a linear projection of this stack, so it is normal by [F1]; thus $(X,Y)$ is jointly Gaussian. Since $q$ is independent of $U$ and centered, $\mathrm{Cov}(X)=F\mathrm{Cov}(U)F^T+\mathrm{Cov}(q)=FP{F}^T+Q=P^{-}$. Since $Y=HX+r$ and $r$ is independent of $X$ and centered, $\mathbb{E}X=Fm$, $\mathbb{E}Y=HFm$, $\mathrm{Cov}(X,Y)=P^{-}{H}^T$, and $\mathrm{Cov}(Y)=H{P}^{-}{H}^T+R=S$. Hence the block covariance is the one displayed in the Statement and is positive semidefinite as a covariance matrix. Also $P^{-}⪰0$, since $u^T{P}^{-}u=(F^{T}u{)}^{T}P(F^{T}u)+u^{T}Qu\ge 0$ for every $u$.

Apply [F4] with $A=P^{-}$, $C=P^{-}{H}^T$, and $B=S$. The observed block is positive definite by hypothesis, and the joint block is positive semidefinite by step 1.1. The conditional mean is $Fm+P^{-}{H}^T{S}^{-1}(y-HFm)=Fm+K(y-HFm)$. Its covariance is $P^{-}-P^{-}{H}^T{S}^{-1}H{P}^{-}=P^{-}-KS{K}^T$, and [F4] makes it positive semidefinite and gives a conditional-law version for every $y$.

Since $S=H{P}^{-}{H}^T+R$ and $K=P^{-}{H}^T{S}^{-1}$, symmetry of $P^{-}$ and $S$ gives $KH{P}^{-}=P^{-}{H}^T{K}^T=KS{K}^T$. Expanding the Joseph expression yields $P^{-}-KH{P}^{-}-P^{-}{H}^T{K}^T+K(H{P}^{-}{H}^T+R)K^T=P^{-}-KS{K}^T$. It is positive semidefinite because [F4] gives that covariance this property; independently, the Joseph expression is a sum of positive-semidefinite matrices. This proves the identity without inverting $P$, $Q$, or $R$.

The Euclidean state and observation spaces are standard Borel. By [F1], the finite path model uses Gaussian laws with the declared positive-semidefinite covariances; [F2] identifies its path law as jointly Gaussian, including when singular. The general existence result [F4] supplies the conditional-law setting, and [F3] fixes the initial filtering kernel as ${\pi }_0(\ast )=N_{d_0}(m_0,P_0)$ on the singleton empty history. This is a probability by [F9] and [F12], and is an RCD of $X_0$ given the trivial history because its event integral is exactly the $X_0$ marginal. The initial covariance is positive semidefinite by hypothesis. At every later step the algorithm is defined on its stated domain: $S_k\succ 0$ has the Cholesky factor required by [F8].

Fix $k\in \{1,\dots ,T\}$ and assume inductively that the algorithm's ${\pi }_{k-1}$ is a filtering kernel with deterministic covariance $P_{k-1\mathrm{\mid }k-1}⪰0$. The vectors $X_{k-1}$ and $Y_{1:k-1}$ are measurable functions of $X_0,q_1,r_1,\dots ,q_{k-1},r_{k-1}$; by [F1,F13], the new pair $(q_k,r_k)$ is independent of that past. Apply [F5] with this past conditional law and push forward by $(x,q,r)\mapsto (F_{k}x+q,H_k(F_{k}x+q)+r)$. For each history $h$, the resulting conditional law of $(X_k,Y_k)$ is the one-step Gaussian block in [F6] with $m=m_{k-1\mathrm{\mid }k-1}(h)$ and $P=P_{k-1\mathrm{\mid }k-1}$. Its predicted covariance is positive semidefinite because $u^T{P}_{k\mathrm{\mid }k-1}u=(F_k^{T}u{)}^T{P}_{k-1\mathrm{\mid }k-1}(F_k^{T}u)+u^T{Q}_{k}u\ge 0$. The positive-definite condition on $S_k$ allows [F7] to condition this block, for every $y_k$, with mean $m_{k\mathrm{\mid }k-1}+P_{k\mathrm{\mid }k-1}{H}_k^T{S}_k^{-1}{v}_k$ and covariance $P_{k\mathrm{\mid }k-1}-P_{k\mathrm{\mid }k-1}{H}_k^T{S}_k^{-1}{H}_k{P}_{k\mathrm{\mid }k-1}$, exactly the algorithm's gain and update. The covariance is positive semidefinite by [F7]. To identify the computed gain, put $B_k=H_k{P}_{k\mathrm{\mid }k-1}$. The algorithm's two triangular solves, applied columnwise, give $L_k{U}_k=B_k$ and $L_k^T{K}_k^T=U_k$. The diagonal entries of $L_k$ and $L_k^T$ are positive, so [F14] gives the unique solution of each column system. Therefore $S_k{K}_k^T=L_k(L_k^T{K}_k^T)=L_k{U}_k=B_k$; since $S_k\succ 0$, this yields $K_k^T=S_k^{-1}{B}_k$, the gain in [F6]. The updated mean is affine in $(h,y_k)$ and the covariance is deterministic, so [F10] makes the Gaussian sections a measurable probability kernel.

Let $J_k(h,\cdot )$ be the conditional law of $(X_k,Y_k)$ given the previous history constructed in step 2.1, and let $M_k(h,\cdot )$ be its observation marginal. The one-step conditional identity [F7] gives $J_k(h,A\times B)={\int }_B{\pi }_k(h,y_k,A)M_k(h,d{y}_k)$ for Borel state $A$ and observation set $B$. Integrating this identity over any Borel set $D$ of previous histories yields $\mathbb{P}(X_k\in A,Y_{1:k-1}\in D,Y_k\in B)={\int }_D{\int }_B{\pi }_k(h,y_k,A)M_k(h,d{y}_k){\mathbb{P}}_{Y_{1:k-1}}(dh)$. Since $M_k$ is the observation marginal of $J_k$, the iterated kernel is the joint law of $(Y_{1:k-1},Y_k)$ integrated against ${\pi }_k$. Thus this is the RCD identity on every rectangle $D\times B$; these rectangles form a pi-system generating the product history sigma-algebra, so [F11] extends the identity to all Borel history events. Therefore ${\pi }_k$ is a filtering kernel by [F3].

The base case is step 1.1, and steps 2.1–3.1 establish the induction step for every $k$, preserving positive semidefiniteness of the updated covariance. For $T=0$ only the base case applies. Hence the stated recursions give a conditional-law version at every history, with no inversion except of positive-definite innovation matrices. Different RCD versions may still differ on observation histories of probability zero.