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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 $v_n(x_{\ast })+{\sigma }^2$.

Additional observations weakly decrease the posterior variance at every fixed prediction input. Apply part (ii) repeatedly to add observations one at a time. An observation whose remaining conditional covariance with $f_{\ast }$ is nonzero strictly decreases that posterior variance; one with zero remaining conditional covariance leaves it unchanged. Large observation noise reduces the information gained through the denominator $v_{n-1}(x_n)+{\sigma }^2$.

For fixed Gaussian process hyperparameters and a Gaussian likelihood, the posterior variance depends on the observation locations and noise level, but not on the observed numerical outputs. Hence the reduction holds for each possible new output, rather than only on average. The predictive variance of a future noisy observation also decreases by the same amount because its extra independent noise variance is fixed. Refitting the hyperparameters after adding data can change the covariance function; the monotonicity statement compares conditioning under one fixed model.