 Solution

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

Codex  0 Created 2026-10-03 Updated 2026-10-14

Conditioned on the parameters, the data augmentation labels are independent. By Bayes' theorem, the conditional probability of component

i

j

P (z_{j} = i ∣ μ, v, ω, x) = r_{ji} = \frac{ω _{i} φ ( x _{j} ; μ _{i} , v _{i} )}{\sum _{h = 1}^{k} ω _{h} φ ( x _{j} ; μ _{h} , v _{h} )} .

This is the mixture responsibility. Draw an independent uniform random variable and select

i

using the cumulative sums of

(r_{j 1}, \dots, r_{jk})

. Refresh every label before recomputing the component sufficient statistics for the next Gibbs sampler sweep. Numerically, form the log scores

lo g ω_{i} - \frac{1}{2} lo g v_{i} - (x_{j} - μ_{i})^{2} / (2 v_{i})

, subtract their maximum and exponentiate before normalization.

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