 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-43/3/a/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-20

Hold all parameters except

μ_{i}

fixed and propose

μ_{i}^{'} = μ_{i} + η

with

η \sim N (0, s_{μ i}^{2})

D_{j}^{'} = D_{j} + ω_{i} {φ (x_{j}; μ_{i}^{'}, v_{i}) - φ (x_{j}; μ_{i}, v_{i})} .

Cancellation of unchanged likelihood and prior factors gives the Metropolis–Hastings acceptance probability

a_{i} = min {1, j = 1 \prod n \frac{D _{j}^{'}}{D _{j}} exp [- \frac{( μ _{i}^{'} ) ^{2} - μ _{i}^{2}}{2 τ ^{2}}]} .

Draw an independent uniform and accept if it is at most

a_{i}

; otherwise retain the old value. Update each component in turn, recomputing the needed mixture contributions. On a logarithmic scale, compare

lo g U

with

min (0, \sum_{j} lo g (D_{j}^{'} / D_{j}) - [(μ_{i}^{'})^{2} - μ_{i}^{2}] / (2 τ^{2}))

to avoid numerical underflow.

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