Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 219 / 2 / d

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- Solution d

Solution

For draw

m

, let

C_{m}

be the observed covariance matrix,

k_{* m} = (k_{θ_{m}} (t_{*}, t_{i}))_{i}

, and

m_{* m} = μ_{m} + k_{* m}^{T} C_{m}^{- 1} (y - μ_{m} 1), v_{* m} = A_{m} - k_{* m}^{T} C_{m}^{- 1} k_{* m} .

The posterior predictive distribution is a mixture of these conditional Gaussians. Its Monte Carlo mean and variance are

m_{*} = \frac{1}{M} \sum_{m} m_{* m}, v_{*} = \frac{1}{M} \sum_{m} (v_{* m} + m_{* m}^{2}) - m_{*}^{2} .

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