Gaussian-process priors and finite posterior laws — Examples

For every finite coefficient list on $T$, the constant kernel has quadratic form $({\sum }_i{a}_i{)}^2\ge 0$. The noise matrix is positive definite because $u^2+uv+v^2=(u+v\mathrm{/}2{)}^2+3v^2\mathrm{/}4>0$ for $(u,v)\mathrm{\ne }(0,0)$. By [F1] there is a cylinder-space GP law $\mu$ and [F2] gives the noise law. Their product probability exists by [F3]; its process and noise coordinates are measurable by [F5] and independent by [F4]. Thus this is a model satisfying the theorem's noise assumption.

The observed and test covariance blocks are $K_{XX}=J_2$, $K_{cX}=(1,1)$, and $K_{cc}=1$. Thus $C=J_2+R=\left(\begin{array}{cc}2& 3\mathrm{/}2\\ 3\mathrm{/}2& 2\end{array}\right)$, which is positive definite since its quadratic form is $2(u+3v\mathrm{/}4{)}^2+7v^2\mathrm{/}8$. Direct multiplication gives $C^{-1}=\left(\begin{array}{cc}8\mathrm{/}7& -6\mathrm{/}7\\ -6\mathrm{/}7& 8\mathrm{/}7\end{array}\right)$.

Since $m_X=(1,-1)$, $m_c=2$, and $y-m_X=(-1,2)$, $C^{-1}(y-m_X)=(-20\mathrm{/}7,22\mathrm{/}7)$ and $K_{cX}{C}^{-1}(y-m_X)=2\mathrm{/}7$. The posterior mean in [F6] is therefore $2+2\mathrm{/}7=16\mathrm{/}7$.

Also $C^{-1}{K}_{Xc}=(2\mathrm{/}7,2\mathrm{/}7{)}^T$, so $K_{cX}{C}^{-1}{K}_{Xc}=4\mathrm{/}7$. The posterior variance is $1-4\mathrm{/}7=3\mathrm{/}7$, proving the displayed conditional-kernel value. This is the value of the all-$y$ kernel version specified in [F6] at $y=(0,1)$; the theorem does not make conditional-law values pointwise unique.

For any finite coefficient list, the constant kernel's quadratic form is $({\sum }_i{a}_i{)}^2\ge 0$. Thus [F1] constructs the cylinder GP and [F2] supplies $N_2(0,I_2)$. Their product probability exists by [F3]; its process and noise random elements are measurable by [F5] and independent by [F4].

The observation covariance is $K_{XX}=J_2$, which has rank one, while $C=J_2+I_2=\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)$ has quadratic form $u^2+v^2+(u+v{)}^2>0$ for every nonzero $(u,v)$. Hence it is positive definite and $C^{-1}=\frac{1}{3}\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)$. For the repeated test list, $K_{\ast X}=K_{\ast \ast }=J_2$. Since $C$ is positive definite, the hypotheses of [F6] hold.

Multiplication gives $K_{\ast X}{C}^{-1}y=(1\mathrm{/}3,1\mathrm{/}3{)}^T$. Also $C^{-1}{J}_2=\frac{1}{3}{J}_2$, so $K_{\ast X}{C}^{-1}{K}_{X\ast }=\frac{2}{3}{J}_2$ and the conditional covariance is $J_2-\frac{2}{3}{J}_2=\frac{1}{3}{J}_2$. Its determinant is zero and its rank is one. By [F6] this is a valid singular Gaussian conditional law.

The prior observation covariance and conditional predictive covariance are both singular, whereas the matrix inverted by the formula is the positive-definite observed covariance $C$. This verifies why singular prior and predictive blocks are permitted under the stated condition.

For any finite list of indices, group coefficients at repeated sites. The equality kernel's quadratic form is the sum, over distinct sites $t$, of $({\sum }_{i:t_i=t}{a}_i{)}^2$, hence is nonnegative. By [F1] the canonical GP law exists and [F2] supplies the $N(0,1)$ noise law. Their product probability exists by [F3]; its process and noise random elements are measurable by [F5] and independent by [F4].

Here $K_{XX}=1$, $K_{\ast \ast }=1$, $K_{\ast X}=K_{X\ast }=0$, and the noise covariance is $R=1$. Thus $C=K_{XX}+R=2>0$, so [F6] applies.

The formula in [F6] gives posterior mean $0+0\cdot 2^{-1}(y-0)=0$ and variance $1-0\cdot 2^{-1}\cdot 0=1$ for every $y\in \mathbb{R}$. Hence the posterior version at the unobserved independent index is its original $N(0,1)$ law.