Past exam of the options I course of the University of Oxford / 2013 / O / 2 / a

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- Solution a

Solution

Let

s = \sum_{j = 1}^{n} x_{j}

. The Bernoulli distribution likelihood is proportional to

θ^{s} (1 - θ)^{n - s}

. Multiplication by the beta distribution prior gives

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

Normalizing with the beta function therefore gives

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 )} .

The parameters remain positive. This is Beta-binomial conjugacy: successes add to the first shape parameter and failures to the second.

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