Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 216 / 1 / b

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- Solution b

Solution

The posterior mean of the number infected is

E [\sum_{i \in V} X_{i} mi dd l e ∣ V_{2}] = \sum_{i \in V} P (X_{i} = 1 ∣ V_{2}) .

Run sum-product belief propagation on the tree to compute every exact one-vertex posterior marginal. Summing their probabilities of state one gives the requested mean. Messages have fixed size and every directed edge is processed once, so the exact computation costs

O (∣ V ∣) .

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