Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 3 / c / ii

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- Solution ii

Solution

For component

m

, stack its

N_{m}

assigned observations as

y_{1}^{(m)}, \dots, y_{N_{m}}^{(m)} \in R^{n}

. The prior is

f_{m} \sim N_{n} (0, K)

and each assigned vector is conditionally

N_{n} (f_{m}, σ^{2} I_{n})

. Normal-normal conjugacy gives

f_{m} ∣ y, c, π \sim N_{n} (μ_{m}, V_{m}),

where

V_{m} = (K^{- 1} + \frac{N _{m}}{σ ^{2}} I_{n})^{- 1}, μ_{m} = V_{m} \frac{1}{σ ^{2}} \sum_{r = 1}^{N_{m}} y_{r}^{(m)} .

If

N_{m} = 0

, this reduces to the prior

N_{n} (0, K)

.

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