Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 219 / 2 / b

Solution

Treat the simulated pairs

(x_{i}, m_{i})

as samples from the prior

P (x, m)

\overset{m}{ˉ} ≃ \frac{\sum _{i = 1}^{K} m _{i} P ( d ∣ x _{i} )}{\sum _{j = 1}^{K} P ( d ∣ x _{j} )} = \sum_{i = 1}^{K} m_{i} w_{i},

w_{i} = \frac{exp [ - ( d - x _{i} ) ^{2} / ( 2 σ ^{2} )]}{\sum _{j = 1}^{K} exp [ - ( d - x _{j} ) ^{2} / ( 2 σ ^{2} )]} .

The common Gaussian normalizing constant cancels.

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