 Solution

ID: past-exam-of-the-advanced-topics-in-statistical-machine-learning-course-of-the-university-of-oxford/2025/3/b/iii/solution

f_{*} ∣ y, θ \sim N (m_{n} (x_{*}), v_{n} (x_{*})),

where

m_{n} (x_{*}) = k_{*}^{T} C^{- 1} y, v_{n} (x_{*}) = k_{**} - k_{*}^{T} C^{- 1} k_{*} .

For a direct justification, form

R = f_{*} - k_{*}^{T} C^{- 1} y

. It is jointly Gaussian with

y

, has zero mean, and satisfies

Cov (R, y) = k_{*}^{T} - k_{*}^{T} C^{- 1} C = 0, Var (R) = k_{**} - k_{*}^{T} C^{- 1} k_{*} .

R

independent of

y

. Conditional on

y

, only the displayed affine shift remains, proving the result. This is the latent-function posterior variance; a new observation

y_{*} = f_{*} + σ ϵ_{*}

with fresh independent noise has predictive variance

v_{n} (x_{*}) + σ^{2}

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