Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 40 / 5 / ii

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- Solution ii

Solution

Use the standard interpretation that the counts are conditionally independent given the parameters, with mutually independent stated priors. Let

a_{2} = α_{2}

,

b_{2} = β_{2}

, and define

T = \sum_{i, j, t} x_{ij t}, R = \sum_{i, t} x_{i 2 t}, C = \sum_{i, j} x_{ij 2}, E = I (1 + e^{a_{2}}) (1 + e^{b_{2}}) .

Under the reference constraints, the Poisson distribution likelihood, up to data-only factors, is

L (θ, a_{2}, b_{2}) = θ^{T} exp {R a_{2} + C b_{2} - θE} .

Multiplying by the shape-rate gamma prior gives the Poisson-gamma conjugacy with unequal exposures

θ ∣ a_{2}, b_{2}, x \sim Γ (a + T, b + I (1 + e^{a_{2}}) (1 + e^{b_{2}})) .

Here

b

is a rate, matching the probability density function convention on the PDF's first page. The factor

I

represents the number of replicates per cell, not the total count.

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