Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 4 / 17H / d

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- Solution d

Solution

For the weighted loss, the Bayes estimator under parameter-weighted squared error is the mean of the posterior after multiplication by

θ^{α - 1} (1 - θ)^{β - 1} .

Since the original posterior density is proportional to

θ^{X} (1 - θ)^{n - X},

the reweighted density is

Beta (X + α, n - X + β)

. Its mean gives

θ = \frac{X + α}{n + α + β} .

Writing this in the required form,

θ = w θ + (1 - w) θ_{0}, w = \frac{n}{n + α + β}, θ_{0} = \frac{α}{α + β} .

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