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

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- Solution a

Solution

The measurement model has the Gaussian likelihood

P (d ∣ x) = \frac{1}{2 π σ ^{2}} exp [- \frac{( d - x ) ^{2}}{2 σ ^{2}}] .

Marginalizing the latent angular momentum gives the normalized posterior distribution

P (m ∣ d) = \frac{\int P ( d ∣ x ) P ( x , m ) d x}{\iint P ( d ∣ x ) P ( x , m ) d x d m} .

Its posterior mean is

\overset{m}{ˉ} = \frac{\iint m P ( d ∣ x ) P ( x , m ) d x d m}{\iint P ( d ∣ x ) P ( x , m ) d x d m} .

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