 Solution

ID: past-exam-of-the-mathematical-and-computational-biology-course-of-the-university-of-cambridge/2025/paper-1/6/d/solution

For fixed prior parameters, divide numerator and denominator by

n

E [p ∣ x] = \frac{1 + a / n}{S / n + ( a + b ) / n} ⟶ \frac{1}{E X} = p_{0},

under independent sampling from a geometric distribution with true success probability

p_{0} > 0

S / n \to 1/ p_{0}

. In a large finite sample the posterior mean is close to

n / S = 1/ \overset{x}{ˉ}

, the maximum-likelihood estimate. The fixed prior's influence vanishes, and the estimate is consistent.

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