Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 207 / 2 / b

Table of contents
- Solution b

Solution

Let

P (t) = e^{Qt}

be the transition probability matrix of the four-state continuous-time Markov chain, and put

N = {S, R}

for a negative test and

P = {E, I}

for a positive test. Starting susceptible at time zero, the likelihood contribution is

\sum_{a \in N} \sum_{b \in P} \sum_{c \in N} P_{S a} (1) P_{ab} (1) P_{b c} (1) .

Equivalently, with indicator diagonal matrices

D_{N}, D_{P}

and the susceptible basis vector

e_{S}

it is

e_{S}^{T} P (1) D_{N} P (1) D_{P} P (1) D_{N} 1 .

This sums over every hidden state sequence compatible with the three test results.

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