Past exam of the statistics course of the University of Oxford / 2017 / 3 / a / ii

Table of contents
- Solution ii

Solution

Conditional on

θ

, the future Bernoulli random variable is independent of the observed statistical sample and has success probability

θ

. Its posterior predictive probability is therefore the posterior mean of

θ

. Put

A = a + s

and

B = b + n - s

. Then

P (X_{n + 1} = 1 ∣ x) = \int_{0}^{1} θ π (θ ∣ x) d θ = \frac{B ( A + 1 , B )}{B ( A , B )} = \frac{Γ ( A + 1 ) Γ ( A + B )}{Γ ( A ) Γ ( A + B + 1 )} = \frac{A}{A + B},

using the beta-gamma identity and the Gamma function recurrence. Hence

P (X_{n + 1} = 1 ∣ X_{1}, \dots, X_{n}) = \frac{a + \sum _{i} X _{i}}{a + b + n} .

The whole posterior predictive distribution of

X_{n + 1}

is the Bernoulli distribution with this success probability.

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