Note on Observation Noise

In practice, it is typical that the observed outputs are contaminated with independent Gaussian noise, so that

\[y_i = f(\mathbf{x}_i) + \epsilon_i, \quad \epsilon_i \sim \mathcal{N}(0, \sigma_n^2)\]

Using the block notation \(\Sigma_{ij}\) for covariance matrices, where \(\Sigma_{11}\) denotes the covariance of the training outputs with themselves, the presence of noise modifies only this block:

\[\Sigma_{11} \;\;\longrightarrow\;\; \Sigma_{11} + \sigma_n^2 I\]

All posterior and marginal-likelihood expressions follow from this replacement. This convention applies generally, including cases with derivatives or multi-output structures.

Noisy Joint Distribution

(1)\[\begin{split}\begin{pmatrix} \mathbf{y} \\ f(\mathbf{X}_*) \end{pmatrix} \sim \mathcal{N}\left( \mathbf{0}, \begin{pmatrix} \Sigma_{11} + \sigma_n^2 I & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{pmatrix} \right)\end{split}\]

Noisy Posterior

(2)\[\begin{split}p(f(\mathbf{X}_*) \mid \mathbf{y}) &= \mathcal{N}(\boldsymbol{\mu}_*, \boldsymbol{\Sigma}_*) \\ \boldsymbol{\mu}_* &= \Sigma_{21} \left(\Sigma_{11} + \sigma_n^2 I\right)^{-1} \mathbf{y}, \\ \boldsymbol{\Sigma}_* &= \Sigma_{22} - \Sigma_{21} \left(\Sigma_{11} + \sigma_n^2 I\right)^{-1} \Sigma_{12}\end{split}\]

Noisy Marginal Log-Likelihood

(3)\[\log p(\mathbf{y} \mid \mathbf{X}, \boldsymbol{\psi}, \sigma_n^2) = -\tfrac{1}{2}\mathbf{y}^\top\!\left(\Sigma_{11} + \sigma_n^2 I\right)^{-1}\mathbf{y} -\tfrac{1}{2}\log\!\left|\Sigma_{11} + \sigma_n^2 I\right| -\tfrac{n}{2}\log(2\pi)\]

Generalization

For augmented problems such as derivative-enhanced GPs or multi-output GPs, the same substitution rule applies. If the training covariance block is, for example, \(\Sigma^{G}_{11}\) or \(\Sigma^{H}_{11}\) of size \(M \times M\), then

\[\Sigma^{G}_{11} \;\;\longrightarrow\;\; \Sigma^{G}_{11} + \sigma_n^2 I_{M}, \qquad \Sigma^{H}_{11} \;\;\longrightarrow\;\; \Sigma^{H}_{11} + \sigma_n^2 I_{M}\]

All posterior and marginal-likelihood expressions remain valid under this substitution.

Heteroscedastic Noise

If distinct observation types have different noise levels (e.g., function values versus gradients, or different outputs), the diagonal correction is replaced with a block-diagonal noise matrix

\[N = \operatorname{diag}(\underbrace{\sigma_{f}^2 I_{n}}_{\text{function block}},\; \underbrace{\sigma_{g}^2 I_{n d}}_{\text{gradient block}},\; \ldots)\]

and the substitution becomes \(\Sigma_{11} \to \Sigma_{11} + N\).

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