Past exam of the advanced topics in statistical machine learning course of the University of Oxford / 2025 / 3 / b

Gaussian process regression supplies a predictive distribution and uncertainty estimates. A Gaussian process regression posterior gives both a posterior mean and a posterior variance, allowing credible intervals for the latent function and a posterior predictive distribution for future noisy observations. The kernel ridge regression estimator by itself specifies a point estimate.

With the same positive-semidefinite kernel and matching regularization, the Gaussian process regression posterior mean equals the kernel ridge regression prediction. Thus the main additional benefit is the explicit probability distribution of predictions, conditional on the chosen Gaussian process model and its hyperparameters, rather than an automatic improvement in point-prediction accuracy.

Condition throughout on the training inputs and $x_{\ast }$. Interpret $\theta$ here as the fixed covariance function and noise hyperparameters. Assume observation errors ${ϵ}_i$ are independent standard normal random variables, independent of the Gaussian process $f$, and writeThe independent-error assumption is the standard Gaussian process regression interpretation of the observation model; the single-observation marginal noise law alone would not specify cross-observation noise correlations. We take ${\sigma }^2>0$ so that $C$ is a positive-definite matrix. More generally the inverse formulas below apply whenever the indicated noisy covariance matrices are invertible.

The Gaussian process prior makes $(f_{\ast },\mathbf{f})$ a Gaussian random vector. Adding the independent Gaussian noise vector to $\mathbf{f}$ preserves a multivariate normal distribution. Its mean is zero, its observation covariance matrix is $C$, and $\mathrm{Cov}(f_{\ast },y_i)=k(x_{\ast },x_i)$ because the noise is independent of $f_{\ast }$. ThereforeA degenerate latent variance is allowed; the displayed multivariate normal distribution then denotes a Gaussian law even if it has no full-dimensional probability density.

The conditional multivariate normal distribution giveswhereFor a direct justification, form $R=f_{\ast }-k_{\ast }^T{C}^{-1}\mathbf{y}$. It is jointly Gaussian with $\mathbf{y}$, has zero mean, and satisfiesThe independence of uncorrelated jointly normal variables therefore makes $R$ independent of $\mathbf{y}$. Conditional on $\mathbf{y}$, only the displayed affine shift remains, proving the result. This is the latent-function posterior variance; a new observation $y_{\ast }=f_{\ast }+\sigma {ϵ}_{\ast }$ with fresh independent noise has predictive variance $v_n(x_{\ast })+{\sigma }^2$.