 Solution

ID: past-exam-of-the-statistics-course-of-the-university-of-oxford/2017/3/a/i/solution

Let

s = \sum_{i} x_{i}

θ^{s} (1 - θ)^{n - s}

π (θ ∣ x) \propto θ^{a + s - 1} (1 - θ)^{b + n - s - 1}, 0 < θ < 1.

Recognizing the normalized beta distribution,

θ ∣ x \sim Beta (a + s, b + n - s), π (θ ∣ x) = \frac{θ ^{a + s - 1} ( 1 - θ ) ^{b + n - s - 1}}{B ( a + s , b + n - s )} .

This is Beta-binomial conjugacy; the same update applies whether the data are recorded as the individual Bernoulli trials or their success count.

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