 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-36/2/g/solution

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

The WinBUGS model uses a finite truncation of the reciprocal prior. The numbered lines have the following roles:

Line (1) forms $p_{j} = (1/ j) / H_{5000}$ , the normalized prior probability vector on the integer labels $1, \dots, 5000$ . The deterministic sum may be declared after its use because the model is declarative.
Line (2) assigns $N$ that categorical distribution. The following observation model treats the supplied value $y = 100$ as observed data, rather than as an unknown parameter.
Line (3) forms observation probabilities $p [j] = 1/ N$ for $j \leq N$ and zero for $j > N$ . For integer $N, j$ , the small positive offset in step(N-j+0.01) includes the endpoint $j = N$ without depending on the convention at zero. There are exactly $N$ nonzero entries, so this vector sums to one.

Combining that categorical distribution observation likelihood with the truncated reciprocal prior produces

p (N ∣ y) \propto N^{- 2} 1_{{y \leq N \leq 5000}} .

The Markov chain Monte Carlo output approximates this posterior. The actual PDF separates the observation line from the preceding comment; it must remain an active stochastic statement.

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