 Solution

ID: past-exam-of-the-statistics-course-of-the-university-of-oxford/2017/3/c/iii/solution

lo g f (x ∣ θ) = x lo g θ + (1 - x) lo g (1 - θ), \partial_{θ} lo g f (x ∣ θ) = \frac{x - θ}{θ ( 1 - θ )} .

θ (1 - θ)

I (θ) = \frac{1}{θ ( 1 - θ )}, π_{J} (θ) \propto θ^{- 1/2} (1 - θ)^{- 1/2} .

B (1/2, 1/2) = π

. Therefore

π_{J} (θ) = \frac{1}{π θ ( 1 - θ )}, 0 < θ < 1; θ \sim Beta (1/2, 1/2) .

This Jeffreys prior for a Bernoulli parameter is proper despite its integrable endpoint singularities. It is the arcsine distribution. It replaces the specified

a, b > 1

prior distribution in part (a) when the Jeffreys prior rule is chosen; it is not one of those restricted priors.

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