Three different “flat” Poisson priors

Topic: Exponential families and conjugacy · more questions on this topic
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A Poisson prior written p(\theta)\propto\theta^{\alpha}e^{-\beta\theta} is \operatorname{Gamma}(\alpha+1,\beta) in the shape–rate convention: the power of \theta is one less than the shape. For an iid sample with sum S and size n, the likelihood contributes \theta^{S}e^{-n\theta}. Derive the posterior Gamma shape produced by each of p(\theta)\propto1, p(\theta)\propto\theta^{-1/2}, and p(\theta)\propto1/\theta. State which is Jeffreys’ prior and which is scale-invariant.

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