Gaussian-process priors and finite posterior laws

By [F4] and [F5], $\mathbb{Q}$ is a countable dense subset of $\mathbb{R}$, so [F2] makes the usual real line separable. Its usual metric is complete by [F3]. Thus the usual topology is Polish by [F1].

The identity map from $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ to the usual Polish real line is a bijection and preserves exactly the Borel sets. The standard-Borel definition [F6] therefore applies.

For each nonempty finite $F\subseteq T$, use AC to choose an enumeration $e_F:\{1,\dots ,n\}\to F$. Set $a_i=m(e_F(i))$ and ${\mathrm{\Sigma }}_{ij}=k(e_F(i),e_F(j))$. Positive semidefiniteness of $k$ makes $\mathrm{\Sigma }$ positive semidefinite; [F1] supplies its positive square root and [F2] gives the possibly singular law $N_n(a,\mathrm{\Sigma })$. Push this law to ${\mathbb{R}}^F$ by relabelling coordinate $i$ as $e_F(i)$. A different enumeration changes both the mean and covariance by the same permutation; [F3] shows that the relabelled law is unchanged. For $F=\mathrm{\varnothing }$, take the unique law on the singleton empty product.

If $F\subseteq G$ are finite, the characteristic function of the restriction pushforward of the $G$-law at $u\in {\mathbb{R}}^F$ is $\exp \text{\hspace{0.17em}⁣}(i{\sum }_{t\in F}{u}_{t}m(t)-\frac{1}{2}{\sum }_{s,t\in F}{u}_s{u}_{t}k(s,t))\mathrm{.}$ This is also the characteristic function of the $F$-law. By [F3] the two pushforward laws agree, so the family constructed in step 1.1 is consistent in the sense of [F4]. The empty marginal is the singleton law.

Each coordinate space $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ is standard Borel by [F5]. The consistent family from step 2.1 therefore satisfies [F6], which gives a unique probability measure $\mu$ on ${\mathcal{C}}_T$ with exactly these finite-coordinate marginals.

Define $X_t(x)=x(t)$ on $({\mathbb{R}}^T,{\mathcal{C}}_T,\mu )$. By [F7] these coordinates are measurable, and step 3.1 gives each finite evaluation vector the stated multivariate normal law. Every finite linear combination is a scalar projection of such a vector, hence is normal by [F2]. Thus [F8] makes $X$ a Gaussian process. Its one- and two-coordinate laws give mean $m$ and covariance $k$, including zero-variance coordinates and singular finite covariance matrices.

The extension in step 3.1 is unique on ${\mathcal{C}}_T$. It supplies no topology or probability law on a path-continuous or Banach-valued function space, so no such regularity or tightness conclusion is included.

By [F1], the concatenated finite evaluation vector $U=(f_{X_{\ast }},f_X)$ is multivariate normal with mean $(m_{\ast },m_X)$ and covariance $\left(\begin{array}{cc}{K}_{\ast \ast }& K_{\ast X}\\ K_{X\ast }& K_{XX}\end{array}\right)\mathrm{.}$ This remains valid when either list has repetitions.

The vector $ϵ$ is multivariate normal with mean zero and covariance $R$. Each coordinate of $U$ is a measurable function of the process coordinates, so $\sigma (U)\subseteq \sigma (f)$. By [F2] and the independence hypothesis, $\sigma (ϵ)$ is independent of $\sigma (f)$ and hence of $\sigma (U)$. Applying [F3] to $U$ and $ϵ$ shows that $Z=(f_{X_{\ast }},f_X,ϵ)$ is jointly multivariate normal with block-diagonal covariance between $U$ and $ϵ$.

Apply the linear map $(u,v,e)\mapsto (u,v+e)$ to $Z$. Every scalar projection of the result is a scalar projection of $Z$, hence is normal by [F4]; its mean is $(m_{\ast },m_X)$. The Gaussian vectors have finite second moments, so independence and $Eϵ=0$ give $\mathrm{Cov}(f_{X_{\ast }},ϵ)=0$ and $\mathrm{Cov}(Y)=K_{XX}+R=C$. By step 1.1, $\mathrm{Cov}(f_{X_{\ast }},f_X)=K_{\ast X}$, so $(f_{X_{\ast }},Y)$ is jointly Gaussian with covariance $\left(\begin{array}{cc}{K}_{\ast \ast }& K_{\ast X}\\ K_{X\ast }& C\end{array}\right)\mathrm{.}$

Apply [F5] with $A=K_{\ast \ast }$, cross-covariance $K_{\ast X}$, and observed covariance $B=C\succ 0$. The result is a probability-kernel version for every $y$, with the displayed mean and Schur-complement covariance; [F5] also proves this covariance is positive semidefinite even if singular.

If ${\eta }_{\ast }$ is present, it is jointly independent of the process and $ϵ$ by hypothesis. By [F3], the stack $(f_{X_{\ast }},f_X,ϵ,{\eta }_{\ast })$ is jointly Gaussian. The linear map $(u,v,e,h)\mapsto (u+h,v+e)$ has mean $(m_{\ast },m_X)$, cross-covariance $K_{\ast X}$, observed covariance $C$, and prediction covariance $K_{\ast \ast }+R_{\ast }$. Applying [F5] to this pair gives the stated noisy-prediction law.

By [F6], write $C=L{L}^T$. For any right-hand side $b$, solve $Lz=b$ by forward substitution and $L^{T}u=z$ by backward substitution; [F8] gives both unique solutions. Then $Cu=L(L^{T}u)=Lz=b$, so $u=C^{-1}b$. Applying this to $b=y-m_X$ and, column by column, to $K_{X\ast }$ evaluates both formula terms without forming the inverse.