Observations x 1 , x 2 , ... , x n are independently distributed given parameters

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Observations x1, x2, ... , xn are independently distributed given parameters θ1, θ2, ... ,θn according to the Poisson distribution p(xi ∣θ) = θixi exp(-0i)/xi. The prior distribution for θ is constructed hierarchically. First, the θis are assumed to be independently identically distributed given a hyperparameter ϕ according to the exponential distribution p(θi∣ϕ) = ϕ exp(-ϕθi) for θ≥ 0 and then ϕ is given the improper uniform prior p (ϕ) ∝ 1 for ϕ ≥ 0. Provided that nX̅ > 1, prove that the posterior distribution of z = 1 / (1 + ϕ) has the beta form 

p(z|x) x zx-2(1-z)". z).

Thereby show that the posterior means of the θi are shrunk by a factor (nX̅ - 1)/(nX̅ + n) relative to the usual classical procedure which estimates each of the θi by X̅i

What happens if nX̅ ≤ 1?

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