Past exam of the statistics course of the University of Oxford / 2015 / 3 / a

Solution

Let

p_{0} = Pr (θ \in Θ_{0})

and

p_{1} = 1 - p_{0}

, assuming both are positive. The prior odds are

α = p_{0} / p_{1}

. Given data

x

O_{01} (x) = \frac{P r ( H _{0} ∣ x )}{P r ( H _{1} ∣ x )} .

With conditional joint likelihood

L (θ; x) = \prod_{i} f (x_{i} ∣ θ)

m_{0} (x) = \frac{1}{p _{0}} \int_{Θ_{0}} L (θ; x) π (d θ), m_{1} (x) = \frac{1}{p _{1}} \int_{Θ ∖ Θ_{0}} L (θ; x) π (d θ) .

H_{0}

relative to

H_{1}

B = m_{0} (x) / m_{1} (x)

p_{0}, p_{1}

O_{01} (x) = \frac{p _{0} m _{0} ( x )}{p _{1} m _{1} ( x )} = α B .

Pr (H_{0} ∣ x) = \frac{α B}{1 + α B} .

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